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Evaluation of Fundamental Viscoelastic Functions by a Nonlinear Viscoelastic Model.

INTRODUCTION

The extended application of polymers and polymer matrix composites as engineering materials in aerospace, automotive industry, and civil engineering structures demands for their correct characterization with regard to the time-dependent properties, that is, long-term dimensional stability and strength, as a consequence of the viscoelasticity of the polymeric matrix [1,2].

Given that the viscoelastic material behavior needs to be characterized in a very broad range of time (or frequency), it is necessary to combine experiments at certain time (frequency) regions and temperatures, applying time-temperature-superposition (TTS) principle. Following this principle, a property measured at short times and high temperatures should be the same as that for longer times and lower temperatures [3, 4], The relationships between the various viscoelastic functions can be expressed by Fourier or Laplace transform. The most important viscoelastic functions are the stress-relaxation and creep moduli in the time domain, and the complex moduli (frequency domain). Performing all the experiments that define the material behavior is not always feasible; therefore, the interconversion functions between the different moduli are an important task in viscoelasticity. Modeling the viscoelastic behavior of polymeric materials over a wide time-frequency and temperature range is always an interesting topic [3, 5]. This analysis can be carried out within the frame of various viscoelastic models, such as standard linear solid, generalized Maxwelll or generalized Kelvin-Voigt model [3]. Regarding linear viscoelastic materials, it is relatively simple to apply a number of methods for the solution of equations that relate the different viscoelastic functions [6-9], in cases where the interconversion is based on the type of loading (axial, shear, or bulk). However, when the interconversion is related to the test type, that is, from static to dynamic moduli, the exact solution of the corresponding integral equations is not always possible. In those cases, the use of the generalized models of Maxwell or Kelvin can be an alternative interconversion procedure.

It must be mentioned, however, that the classical linear theory of viscoelasticity [10, 11], which adequately describes the creep, relaxation, and loading-rate dependence of polymers, can be presented by two major forms: hereditary integrals or differential forms. This presentation, nevertheless, is limited to narrow loading rate and temperature regimes, while polymeric materials demonstrate a nonlinear response at relatively small strains.

The most general multiple integral constitutive relation for a nonlinear viscoelastic material is given by Green-Rivlin theory [12]. The complexity of this model and the large amount of experimental data required to determine the material parameters resulted in a very limited use. Significant works related to more applicable models for nonlinear viscoelasticity have been developed [13-15]. A number of physical and semiempirical single integral constitutive relations have also been proposed [16], while the modified superposition principle was first introduced by Leaderman in 1940 [15]. Progress has been made in developing mathematical models for the small strain regime under a specific narrow spectrum of strain rates [17], while much less progress has been made for multiaxial finite deformation response under a wide range of strain rates and temperatures, from a continuum point of view [18].

Given the importance of the nonlinear viscoelasticity, which very often can be studied in combination with viscoplasticity, both tasks have been the subject of a number of works [19-28].

To this trend, the nonlinear viscoelastic/viscoplastic response has been elaborated by the transient model [29] and by the theory introduced by Drozdov [27]. In our previous works [23, 24], a new theory describing the nonlinear viscoelastic/viscoplastic response of polymeric systems--under monotonie loading, stress relaxation, and creep--has been introduced on the basis of Drozdov's work [27]. According to this model, a viscoelastic medium is treated as a network of long chains connected by physical crosslinks and entanglements (elastic links) which break and reform. The viscoelastic response is related to the rearrangement of polymer chains, that is, detachment of active chains or attachment of dangling chains to temporary junctions. The rate of detachment of active chains from their junctions was expressed by an Eyring type equation, while the number of active chains could be calculated by a first-order kinetics. A time-dependent strain energy equation could thus be obtained.

As a modification to this treatment, in this work, a fractional derivative that controls the rate of molecular chain detachment from their junctions has been adopted, leading to a nonlinear constitutive equation in viscoelasticity. As mentioned in Ref. 30, fractional derivative viscoelastic models accurately represent the essentially frequency independent response of materials in the low-frequency range, while they have also been capable of matching experimental data in wider frequency ranges.

The proposed description was proved to be more flexible, given that it contains a relaxation function that has a more global character, appropriate for cases where heavy tails are expected. The proposed time-dependent constitutive equation has been employed to simulate the experimental data of a compliance master curve, in a quite broad time scale. Hereafter, the same set of model parameters was used to predict the experimental dynamic moduli in an equivalent frequency scale. This procedure has been applied for two different polymeric structures, and the simulation results seem to be stimulating for further establishing a method of interconversion of viscoelastic functions.

CONSTITUTIVE ANALYSIS

Nonlinear Viscoelastic Model

According to the transient network theory and the viscoplastic model introduced in Ref. 27, polymeric chains are distinguished into permanent and transient, with the latter being rearranged upon deformation. Therefore, the number of chains (active) which contribute to the polymeric network is evolved with time. In Fig. 1, a schematic presentation of the molecular chains rearrangement upon deformation is depicted.

In this analysis, all calculations have been performed for compressible material, providing thus a more generalized approach than that of an incompressible solid previously examined.

Summarizing the basic assumptions of Drozdov's analysis, we have the rate of separation of active chains from their junctions to be given by an Eyring type equation:

F(u) = [gamma] exp(- [mu]) (1)

Where u is the activation energy and [gamma] is the attempt rate [27], which determines the kinetics of molecular chain rearrangement and is correlated with the time scale of the experiment. In addition, function n(t,[tau],u) was proposed, which expresses the number (per unit volume) of transient chains at time t [greater than or equal to] 0 that have returned into the active state before [tau] [less than or equal to] t and are characterized by an activation energy u. In that way, the memory of the material regarding its strain (stress) prehistory is introduced and this function will be shown to be a key function during the rearrangement procedure. In addition, n(t,0,u) expresses the number of transient chains, active in the reference state, which have not been detached from their junctions until time t, and n(t,t,u) is the number of active transient chains that have activation energy u at time t. The later quantity is expressed as n(t,t,u) = [N.sub.a] f(u), where [N.sub.a] is the number of active transient chains and f(u) expresses the distribution function, which is followed by the activation energy u. According to the random energy model [31], f(u) is defined as

f(u) = [f.sub.0] exp - [- [1/2] [(u/[SIGMA].sup.2])] (u [greater than or equal to] 0), F(h) = 0 (u < 0) (2)

Equation 2 represents a quasi-Gaussian distribution function, with a mean value equal to zero and a standard deviation [SIGMA].

In the work by Drozdov [27], the rate of molecular chain rearrangements has been calculated by first-order kinetics. In the present analysis, the noninteger (fractional derivative) has been adopted. The integer-order derivative is a local operator; therefore, it is not appropriate in cases where heavy tails are expected, and the influence of larger neighborhood cannot be neglected anymore [32]. On the other hand, a fractional derivative is global; it considers this effect, and provides the possibility of different probabilities for the motion forward and backward. Therefore, the evolution of the detachment of active chains from their junctions is based on the study of functions n(t,0,u) and n(t,[tau],u). Starting from the kinetics of the first function, we have

[[partial derivative].sup.a]n/[partial derivative][t.sup.a](t,0,u) = -F(u) n(t,0,u) (3)

where [alpha] is the order of derivative, and 0< [alpha] < 1. The general form of the solution of Eq. 3 is given by

n(t,0,u) = [N.sub.a]f(u) [E.sub.a](-[(F(u) t).sup.a]) (4)

Where [E.sub.[alpha]] is the Mittag-Leffler [33] and is given by the expression

[E.sub.a] (-[(F(u)t).sup.a]) = [[infinity].summation over (n=0)] [(-1).sup.n] [(F(u)t).sup.an]/[GAMMA](na + 1), 0 < a < 1 (5)

with [GAMMA] being the well-known gamma function [33]. In the case of [alpha] = 0.5, the solution of Eq. 4, is given by

[mathematical expression not reproducible] (6)

where erfc is the complementary error function.

Proceeding further the kinetics of function n(t,[tau],u), the rate of chain attachment to the network junctions is given by

[partial derivative]n/[partial derivative][tau](t,[tau],u)=F(u) [N.sub.a] f(u), (for t=[tau]) (7)

Hereafter, we can calculate in a similar way the detachment rate from the chain junctions as

[mathematical expression not reproducible] (8)

The solution of Eq. 8 has the form

[mathematical expression not reproducible] (9)

The integration of Eqs. 3 and 8 has been performed applying the conservation law for the number of active chains, leading to the corresponding solutions given by Eqs. 6 and 9.

Equations 6 and 9 provide the necessary quantities, regarding the number of molecular chains, which actually participate to the materials mechanical response. Proceeding further to the strain energy calculation, the general form of strain energy of an active chain is given by

W = [1/2] [bar.u] [[epsilon].sub.ij] [[epsilon].sub.ij] + [bar.u]v/2(1 - 2v) [[epsilon].sup.2.sb.kk] (10)

Where [bar.u] is the rigidity, v is the Poisson's ratio, and [[epsilon].sub.ij] is the deformation tensor.

By taking the summation of the strain energy of permanent chains and transient chains which have not been rearranged within the time period (0, t), the total strain energy of the equivalent network is as follows:

[mathematical expression not reproducible] (11)

Where [N.sub.p] is the number of permanent chains, n(t,0,u) is defined by Eq. 6, and [partial derivative]n/[partial derivative][tau] (t, [tau], u) is defined by Eq. 9. The first and third term of this equation correspond to the energy of permanent chains and active chains that have not contributed to the rearrangement induced by the imposed strain until time t. The second and fourth term are related with the strain energy of molecular chains that have merged at previous time [tau]. Following Ref. 26, the derivative of W(t) with respect to time gives

[mathematical expression not reproducible] (12)

Where

[mathematical expression not reproducible] (13)

and

[mathematical expression not reproducible] (14)

For isothermal deformation, the Clausius -Duhem inequality reads

Q(t) = dW/dt (t) + [[sigma].sub.ij](t) d[[epsilon].sub.ij]/dt(t) [greater than or equal to] 0 (15)

Where Q is the internal dissipated energy per unit volume. From Eq. 15, taking into account Eqs (12-14), as well as Eqs 6 and 9, and considering that [A.sub.1](t) and [A.sub.2](t) are always positive quantities, we obtain

[mathematical expression not reproducible] (16)

Where E/(1 + v) is equal to the product [bar.[mu]]([N.sub.a] + [N.sub.p]), [[delta].sub.ij] is the Kronecker index and function R(t) is given by

[mathematical expression not reproducible] (17)

Where [kappa] = [N.sub.a]([N.sub.a] + [N.sub.p]) is a constant, and E is the Young's modulus.

According to the above analysis, Eq. 16 is a three-dimensional, time-dependent constitutive equation, in the nonlinear viscoelastic regime.

For a uniaxial experiment, Eq. 16 is given by

[mathematical expression not reproducible] (18)

Where [sigma](t) and [epsilon](t) are the uniaxial stress and strain correspondingly. Constitutive Eq. 18 is a nonlinear, time-dependent, memory integral expression, whereas R(t) is actually a relaxation function.

Dynamic Moduli Calculation

If it is assumed, the imposition of a strain function e.(t) of the general form

[epsilon](i) = [[epsilon].sub.0] [e.sup.i[omega]t] (19)

Where [[epsilon].sub.0] is the strain amplitude and [omega] is the frequency, following constitutive Eq. 18, the complex modulus [E.sup.*] will be given by

[E.sup.*] = [sigma](t)/[epsilon](t) = E'([omega]) + iE'([omega]) (20)

Where E'([omega]) and E"([omega]), are the storage and loss modulus correspondingly, which will be given by

[mathematical expression not reproducible] (21)

MODEL SIMULATION

To check the proposed model's capability of simulating the various types of viscoelastic functions, the following treatment has been applied. As a first step, the experimental data of a creep compliance master curve D(t) have been matched with Eq. 18. To perform this, Eq. 18 together with Eqs 1, 2, and 17 were solved for a constant stress and the variation of strain (compliance) could thus be obtained. All calculations, due to the complexity of involved equations, were performed numerically, using the software Mathematica. The fitting procedure was based on a back analysis treatment, whereas the time/frequency range which could be examined was restricted by our potential in computing time.

The required parameter values are [kappa], [gamma], [SIGMA], E, and v. Regarding Poisson's ratio, in this work, it was assumed that it remains constant with varying time for reasons of simplification. As mentioned in Ref. 34, within the frame of viscoelasticity, this quantity is not only time-dependent, but depends on test modality as well. Therefore, further investigation is required if Poisson's ratio is treated as a viscoelastic function. Given that elastic constants E and v can be known from an independent experiment, only three parameters [kappa], [gamma], and [SIGMA] need to be evaluated. Hereafter, the set of calculated parameters was used to evaluate the material's viscoelastic functions in the frequency domain, that is, storage (E') and loss modulus (E"), with regard to Eq. 21. The experimental data of two different polymeric materials, namely polymethyl-methacrylate (PMMA) [35] and a biodegradable polymer under the commercial name Ecovio [36] were employed. The experimental procedure is described in detail in Refs. 35, 36. Creep compliance and dynamic moduli master curves are experimentally available for both material types, in a wide time and frequency scale. The corresponding master curves are depicted in Figs. 2-6. It is observed from these figures that the transition region in PMMA is quite narrow, extending in a region of three orders of magnitude (Figs. 2-4), while it is much broader for Ecovio (Figs. 5 and 6). In addition, this material type does not exhibit the low- and high time (frequency) plateau region; instead, it appears to have a smoothly increasing slope in both limit regimes, while two main relaxation mechanisms seem to be present. This different behavior of Ecovio can be related with a broad distribution of relaxation (retardation) times, which may be related to the fact that Ecovio is a blend of poly(butylene adipate-terephthalate) copolyester, which is based on nonrenewable resources, and polylactic acid (PLA).

Given that simulation procedure is performed in a time scale of five orders of magnitude or higher, the active chains, which need high activation energy values to participate to the rearrangement, can also be included assuming that the standard deviation [SIGMA] is a function of the time scale. Otherwise, the molecular chains with activation energy higher than that which is covered by the constant [SIGMA] have no possibility to participate to the chain rearrangement procedure, even when the time available is long enough.

Therefore, it is reasonable to consider that [SIGMA], which is the standard deviation of the distributed activation energy (Eq. 2), is evolved with time.

Considering that the number of active chains at time t is given by t [gamma] exp(-u), then the standard deviation by definition will be written as

[mathematical expression not reproducible] (22)

Where [bar.u] is the mean value of the distribution density and is equal to zero. Following this, the number of required parameters is reduced.

Regarding PMMA, it was possible to simulate the creep compliance experimental data by a set of model parameters, as shown in Table 1. It must be noted here that apart from elastic properties E, v of the material, two model parameters, namely, [kappa], [gamma] are required.

The quality of simulation for PMMA creep compliance master curve is shown in Fig. 2, and it can be observed that a time scale of five orders of magnitude was possible to be simulated with a very good approximation. Hereafter, the storage and loss modulus of PMMA were calculated according to Eqs. 21, with the same set of parameters, and the same relaxation function R(t), as evaluated in the simulation of creep compliance. It must be mentioned that the frequency region where a good approximation is achieved, for the loss modulus is narrower than the one for the storage modulus (Figs. 3 and 4). This can be attributed to possible deviations of the experimental results and the application of the time-temperature superposition (TTS) principle. This effect has been thoroughly discussed by Sane and Knauss [37]. and it was deduced that whereas lack of interconvertability is attributed to failure of TTS, a number of reasons are accounted for. Some of these issues are generally related to the nonapplicability of TTS in the glassy domain, or the existence of secondary transition in the glassy state.

Proceeding further to the simulation of the creep compliance experimental data of Ecovio, shown in Fig. 5, it may be observed as already mentioned, that unlike PMMA, the two limit regions (low and high time) appear to have a smooth slope instead of plateau values. The model was proved to be flexible enough to describe this behavior. The simulation results are depicted in Figs. 5-7, in terms of compliance, storage, and loss modulus, respectively. From these plots, it is extracted that compliance and storage modulus could be predicted for a time scale of five orders of magnitude, with a very good approximation. Owing to the limitations of our potential in computing time, it was not possible to perform simulations in a broader time scale. Regarding loss modulus (Fig. 7), it was possible to have a good correlation of experimental data in the main transition frequency region. This material exhibits two main relaxation peaks, due to its composition, as already mentioned.

Summarizing the model presented here, and to show the improvements provided by the fractional derivative, parallel simulation has been performed with the integer derivative. In Fig. 8, the relaxation function R(t) with varying time is illustrated for both fractional and first-order kinetics. As shown in this plot, after an initial time period of coincidence, essential differences between the two approaches are obvious. This is further shown in Fig. 9, where the simulated creep compliance of PMMA with both methods is comparatively depicted. The results initially are in good agreement, whereas at longer times, high deviations appear for the integer derivative approach, supporting this way the fact that fractional derivative remains important for longer times, compared to that by the first-order kinetics.

CONCLUSIONS

In this work, a time-dependent constitutive equation has been derived, on the basis of transient polymer network, presented and analyzed in previous works. Given that viscoelasticity is described by the concept of transient chains, which rearrange upon deformation by either attaching or detaching from their junctions, the kinetics of the detachment of active molecular chains is a crucial procedure for the obtained constitutive equation. In the present analysis, a new assumption, considering fractional derivatives, has been performed resulting to a modified constitutive equation. The proposed equation contains a relaxation function which remains important at longer times. Therefore, it has been further explored the potential of this equation to be employed for the evaluation of the dynamic modulus master curve, once the experimental data of the compliance modulus of a polymeric structure have been simulated. A new approach related with the time dependence of the standard deviation of the activation energy distribution density function is additionally introduced, resulting to a more flexible model simulation. The experimental data (compliance and dynamic moduli master curves) of two different polymers, namely PMMA and Ecovio, were employed to check the model's capability. A satisfactory agreement between experimental data of compliance has been achieved and hereafter the dynamic moduli could be moderately predicted with the same model parameters.

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Gerasimos Spathis, Stylianos Katsourinis, Evagelia Kontou [iD]

Department of Applied Mathematical and Physical Sciences, Section of Mechanics, National Technical University of Athens, Athens GR-15773, Greece

Correspondence to: E. Kontou; email: ekontou@central.ntua.gr

DOI 10.1002/pen.24525

Published online in Wiley Online Library (wileyonlinelibrary.com).

Caption: FIG. 1. Schematic presentation of molecular chain rearrangement of the polymeric network upon the imposition of strain.

Caption: F1G. 2. Creep compliance master curve of PMMA with 1% crosslinking degree, at a reference temperature of 148[degrees]C after Alves et al. [35], Points: experimental data; thick lines: model simulation. [Color figure can be viewed at wileyonlinelibrary.com]

Caption: FIG. 3. Storage modulus master curve of PMMA with 1% crosslinking degree, at a reference temperature of 148[degrees]C after Alves et al. [35], Points: experimental data; thick lines: model simulation. [Color figure can be viewed at wileyonlinelibrary.com]

Caption: FIG. 4. Loss modulus master curve of PMMA with 1% crosslinking degree, at a reference temperature of 148[degrees]C after Alves et al. [35]. Points: experimental data; thick lines: model simulation. [Color figure can be viewed at wileyonlinelibrary.com]

Caption: FIG. 5. Creep compliance master curve of Ecovio at a reference temperature of 65[degrees]C after Georgiopoulos et al. [361. Points: experimental data; thick lines: model simulation. [Color figure can be viewed at wileyonlinelibrary.com]

Caption: FIG. 6. Storage modulus master curve of Ecovio at a reference temperature of 65[degrees]C after Georgiopoulos et al. [36]. Points: experimental data; thick lines: model simulation. [Color figure can be viewed at wileyonlinelibrary.com]

Caption: FIG. 7. Loss modulus master curve of Ecovio at a reference temperature of 65[degrees]C after Georgiopoulos et al. [36], Points: experimental data; thick lines: model simulation. [Color figure can be viewed at wileyonlinelibrary.com]

Caption: FIG. 8. Variation of relaxation function R(t) calculated by first-order and fractional derivative kinetics. [Color figure can be viewed at wileyonlinelibrary.com]

Caption: FIG. 9. Creep compliance master curve of PMMA with 1% crosslinking degree, at a reference temperature of 148[degrees]C after Alves et al. [35]. Points: experimental data; thick lines: comparative plots of first-order and fractional derivative kinetics model simulation. [Color figure can be viewed at wileyonlinelibrary.com]
TABLE 1. Model parameter values.

                                Value

Model parameter        PMMA                Ecovio

E(a)                 150 MPa              180 MPa

K                      0.53                0.38

[GAMMA]                2500           2.5 x [10.sup.-5]
                   ([s.sup.-1])          ([s.sup.-1])

[SIGMA]           [square root of      [square root of
                    2 t [gamma]]         2 t [gamma]]

N                       0.3                 0.3

(a) Modulus values relative to the reference temperature of master
curve
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