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A viscoelastic--plastic constitutive model for uniaxial ratcheting behaviors of polycarbonate.


Polycarbonate (PC) has been extensively utilized in automatic vehicle, civil engineering and aerospace industry due to its excellent mechanical properties [1, 2], The structural components made by PC are often subjected to cyclic loading, a key issue and curial impact to their fatigue life and safe assessment.

Under the stress-controlled asymmetric cyclic loading, i.e., ratcheting, the inelastic deformation will progressively accumulate during the loading process. As reviewed by Ohno [3], Kang [4], Chaboche [5], many works have been conducted on ratcheting behavior of metallic materials both experimentally and theoretically.

Recently, many experimental observations on the cyclic deformation of polymer materials had been done. Shen et al. [6] found that the stress-strain response of uniaxial cyclic tests on epoxy depended on the loading amplitude and loading rate, while a different response was observed for tension and compression cyclic loading. More experimental efforts of cyclic deformation, such as uniaxial/multiaxial ratcheting, fatigue on epoxy were performed by Hu et al. [7], Xia et al. [8), Tao and Xia [9. 10], da Coasta Mattos and de Abreu Martins [11], etc. Chen and Hui [12] and Zhang et al. [13] investigated the uniaxial ratcheting behavior of polytetrafluoroethylene (PTFE) with different loading rate, mean stress and stress amplitude at ambient and elevated temperatures, respectively. In addition, the multiaxial ratcheting of PTFE at room temperature was carried out by Zhang and Chen [14]. Nguyen et al. [15] investigated stress-controlled tension-tension cyclic deformation of high density polyethylene (HDPE). In addition, ratcheting on polymethyl methacrylate (PMMA) [16], vulcanized nature rubber [17, 18], polymer-based anisotropic conductive film [19, 20], were also studied. Meanwhile, the influences of temperature and molecular weight on uniaxial ratcheting behavior of PC were considered by Lu et al. [21] and Xi et al. [22], respectively. Pan et al. [23] demonstrated that the ratcheting deformation of polyetherimide (PEI) would occur both before and after the macroscopic yield, while the ratcheting strain accumulation after yield point is more apparent than that before. More recently, Jiang et al. [24] proposed a test procedure to separate the contributions of viscous recovery and accumulated peimanent deformation under cyclic loading. The test results certify that for PC, previous viscous deformation has significant influence on the following cyclic loading. Both the possible recovery and unrecoverable parts are in same level of magnitude, thus neither of them can be neglected.

Based on the experimental observations, many constitutive models were explored to describe the cyclic deformation of polymer material. Liu et al. [16] proposed a semiempirical model to describe the steady accumulation of ratcheting strain vs. number of cycles, while lacking the capability to reproduce the cyclic stress-strain curves. Zhang et al. [25] studied PTFE using a modified ratcheting model of metal with the consideration of rate-dependent characteristic of polymer. Pan et al. [26] proposed a nonlinear viscoelastic cyclic constitutive model by extending the Schapery model [27] to describe the tension-tension and tension-compression ratcheting behavior of PEI. Xia et al. [28, 29] on epoxy and Nguyen et al. [15] on HDPE, also attempted to establish the constitutive model based on the viscoelastic framework. Thus, a full recovery of the ratcheting strain is eventually unavoidable against the experiment observation. To reasonably describe the ratcheting of polymer, the plastic unrecoverable deformation, as well as the viscorecovery part, has to be considered. A cyclic viscoplastic constitutive model [30] and a cyclic linear viscoelasticity, nonlinear viscoplasticity model based on infinitesimal strain [31] were developed to study the strain-controlled cyclic tensile behaviors on isotactic polypropylene (PP). Obviously, in addition to these factors, such as mean stress, stress amplitude, loading rate, loading path, and loading history, the cyclic viscoelastic-plastic response has to be considered in the constitutive model to reasonably describe the ratcheting of polymer.

A finite viscoelastic-plastic constitutive model was developed by Anand and Ames [32, 33] to model the behavior of amorphous polymeric materials. Based on their framework, a modified viscoelastic-plastic model was proposed in this paper to describe the uniaxial ratcheting behavior of PC under tension-tension cyclic loading. To include the influence of previous deformation on the following loading cycle, we assumed that the viscoelastic resistance is evolving with the local accumulated deformation, rather than the constant assumption in Anand model. The modified model is able to reproduce the evolution tendency of the hysteresis loop with increasing number of cycles. Its capability to predict the influences of mean stress, stress amplitude, stress rate, and peak holding time on the uniaxial ratcheting of PC is also discussed.



The Anand Model

Anand and Ames [32, 33] have proposed a finite deformation viscoelastic-plastic constitutive model to describe the highly nonlinear stress-strain behavior of amorphous glassy polymers. Compared with other constitutive models developed by Haward and Thackray [34], Boyce et al. [35], Arruda and Boyce [36], Wu and Van der Giessen [37], Anand and Gurtin [38], Miehe et al. [39], Fleischhauer et al. [40], the Anand model is not only strictly based on a thermodynamics framework, but also considers the viscoelastic response which is an important inherent characteristic of polymers. Its 1-d rheological model is shown in Figure 1.

The left part of Figure 1 consists of a linear elastic spring and a submodel of Kelvin-Voigt elements in series which describe the inelastic micromechanisms, i.e., viscoelastic and viscoplastic micromechanism. The stretch U is decomposed as U = [U.sup.e] [U.sup.p], where [U.sup.e] and [U.sup.p] are elastic and plastic part, respectively. The stress is expressed as

[[sigma].sub.L] = [E[epsilon].sup.e], with [[epsilon].sup.e] = ln[U.sup.e] (1)

The flow rule, [[??].sup.P] = [D.sup.P][U.sup.P]. Here, [D.sup.P] denoted by the sum of plastic stretches for the overall micromechanisms.

[D.sup.P] = [N.summation over ([alpha]=0)] [D.sup.P([alpha])] = [N.summation over ([alpha]=0)] [v.sup.([alpha])] sign([[sigma].sub.L] - [[sigma].sup.([alpha]).sub.back]) (2)

The [v.sup.([alpha])] representing an effective tensile plastic strain rate for the [alpha] th micromechanism is formulated as a simple power law form

[v.sup.([alpha])] = [V.sub.0] ([absolute value of [[sigma].sub.L] - [[sigma].sup.([alpha]).sub.back]]/[s.sup.([alpha])] - 1/3 [[alpha].sub.P][[sigma].sub.L]).sup.1/[m.sup.([alpha])] (3)

where [v.sub.0] and [[alpha].sub.p] are the reference plastic strain rate and the pressure sensitivity parameter, respectively, and 0 < [m.sup.([alpha])] [less than or equal to] 1 are strain rate sensitivity parameters. The constitutive functions of the backstresses [[sigma].sup.([alpha]).sub.back] and inelastic deformation resistance [s.sup.([alpha])] are reformulated for the viscoelastic and viscoplastic micromechanism, respectively.

For the viscoplastic micromechanism that represents by [alpha] = 0, [[sigma].sup.(0).sub.back] = 0, and [s.sup.(0)] was formulated as a coupled rate-independent form with the internal-state variable free volume [phi].


where {[h.sub.0], [g.sub.0], [s.sup.(0)], b, [[phi]]} are additional material parameters. The initial values of [s.sup.(0)] and [phi] are denoted by [s.sub.i.sup.(0)] and [[phi].sub.i].

For the [alpha]th viscoelastic micromechanism, [[sigma].sup.([alpha]).sub.back] = [[mu].sup.([alpha])] [[(A.sup.([alpha])]).sup.2] - where the evolution equations of stretch-like internal variables [A.sup.(alpha)] are [[??].sup.([alpha])] = [A.sup.([alpha])] [D.sup.P([alpha])] and the back-stress moduli [[mu].sup.([alpha])] are [[??].sup.([alpha])] = [c.sup.([alpha])] (1 - [[mu].sup.([alpha])/[[mu].sup.([alpha]).sat] [??].

Since the model is initially designated to describe the monotonic deformation of glassy polymer, the viscoelastic resistance for the [alpha]th Kelvin-Voigt element is assumed to be constants during whole deformation process. Here is denoted as [s.sup.([alpha])]

[s.sup.([alpha])] = [s.sup.([alpha]).sub.i], [alpha] = 1, ..., N (5)

where [s.sup.([alpha]).sub.i] denote the initial value.

In the right part of Figure 1, a Langevin spring was utilized to describe the postyield hardening behavior under finite deformation. An effective stretch is defined as [bar.[lambda]] = 1/[square root of 3] [square root [U.sup.2] + 2[U.sup.-1]]. The stress is expressed as

[[sigma].sub.R] = [[mu].sub.B] [[U.sup.2] - [U.sup.-1]] with [[mu].sub.B] = [[mu].sub.R] ([[lambda].sub.L]/3[bar.[lambda]]) [[Laplace].sup.-1] ([bar.[lambda]]/[[lambda].sub.L]) (6)

where [[Laplace].sup.-1] is the inverse of the Langevin function. The rubbery modulus [[mu].sub.R] and the network locking stretch [[lambda].sub.L] are two material parameters.

The Anand model has been successfully employed to simulate the time-dependent behavior of PMMA under various loading conditions such as monotonic compression, creep, strain-controlled loading and unloading test, and even the p-h curves of microindentation with/ without time hold. However, there is no stress-controlled asymmetric cyclic loading, i.e., ratcheting been involved.

The Modified Model

For ratcheting, the deformation accumulates successively along the direction of mean stress during the cyclic loading process. Benaarbia et al. [41] proposed a concept of mean slope of the hysteresis loop introduced to characterize the material stiffness during cyclic loading. As to the cyclic test of PC in our previous work [24], the cyclic response significantly depends on the history of its previous deformation. The mean slope evolves with the increasing number of cycles, as well as the shape of hysteresis loop. Thus, different from the assumption that the viscoelastic deformation resistances were not changed during the whole loading process in the Anand model, we assumed that [s.sup.([alpha])]([alpha] = 1, ..., N) in Kelvin-Voigt elements are evolving with the accumulated local inelastic strain. And the evolution functions for [s.sup.([alpha])] are given in the form of

[[??].sup.([alpha])] = [d.sup.([alpha])] (1 - [s.sup.([alpha])]/[s.sup.([alpha]).sub.sat]) [v.sup.([alpha])] (7)

For the [alpha]th element, [s.sup.([alpha]).sub.i] is the initial value of viscoelastic deformation resistance with the same physical meaning of that in Eq. 5, [s.sup.([alpha]).sub.sat] represents the saturation value and [d.sup.([alpha])] is a material parameter. It should be noted that [s.sup.([alpha]).sub.i] < [s.sup.([alpha])] < [s.sup.([alpha]).sub.sat].

Parameter Identification

To simulate the deformation behavior of polymer materials, the constitutive parameters of Anand model that need to be determined are


All parameters can be identified based on the discussion in Anand and Ames [32, 33].

For the modified model, in addition to the above ones, {[s.sup.([alpha]).sub.sat], [d.sup.([alpha])]} are also needed. These parameters can be obtained from a specific cyclic stress-strain curve and corresponding ratcheting strain curve by trail-and-error.


In this section, the capability of the modified model to predict the deformation behavior of PC under various loading conditions is verified. The experimental data are obtained from previous work of Jiang et al. [24]. The used specimens were dumbbell-shape PC planar (PC NOVAREXR 7030R) with a 4 mm thickness and 80 mm X 10 mm gauge area. The loading conditions consist of monotonic tension, tensile creep, and uniaxial stress-controlled tension-tension cyclic loading. The post-yield hardening is not further discussed in this paper because the load level of these ratcheting tests are below the macro-yield strength, and the maximum prescribe tension strain is only ~15% that is far less than the beginning of the strain hardening. A remarkable fact is that since the viscous deformation is in the same level of magnitude as the accumulated unrecoverable plastic part, neither one should be ignored.

In the subsequent discussion, following the work of Anand and Ames [32, 33], the number of the Kelvin-Voigt elements utilized to mimic the micromechanisms of viscoelastic behavior of polymer was chosen to be 3. Meanwhile, the monotonic tension test at 0.0006 [s.sup.-1] loading rate and tensile creep test at 45 MPa were used to determine the material parameters within the Anand model and corresponding results are listed in Table 1. The cyclic loading case of 30 [+ or -] 15 MPa (i.e., mean stress is 30 MPa and stress amplitude is 15 MPa) at 1 MPa/s loading rate was used to determine these parameters in Eq. 7 and the corresponding results are listed in Table 2.

Uniaxial Tension and Creep

To validate its capability to describe the monotonic deformation of PC, the modified model is firstly verified by the strain-controlled/stress-controlled monotonic tension tests as well as the tension creep tests.

Figure 2 shows the experimental and simulated stress-strain curves of uniaxial tension response of PC at two prescribed strain rates, i.e., 0.0006 and 0.012 [s.sup.-1] respectively. Without surprise, the simulation results from the Anand and modified model are in good agreement with those of experiment. The characteristic of strain rate dependence of PC can be well described using the two models. In addition, the influence of stress rate on the tensile response of PC was also reproduced by both models as shown in Figure 3. It is clear that the modified model can reasonably describe the monotonic tension and its rate-dependent behavior of PC.

The tension creep tests of PC at prescribed stress levels of both 45 and 50 MPa are simulated too. The simulation and corresponding experiment results are shown in Figure 4. For the case of 45 MPa in Figure 4a, the simulated curves of creep strain vs. time by both models agrees quit well with that of experiment. When the stress level reaches 50 MPa, almost the yield strength of PC, the experimental creep strain curve increases rapidly after ~60 s (as shown in Fig. 4b) that results in a quick specimen failure which is believed to be induced by the creep damage from the high level of external stress. Without the consideration of creep damage mechanism in both models, their simulation results exhibit only similar tendency but no sudden burst of creep strain.


Tension-Tension Cyclic Loading

To quantitatively evaluate ratcheting behavior of PC, the ratcheting strain [[epsilon].sub.r], i.e., [[epsilon].sub.r] = ([[epsilon].sub.max] + [[epsilon].sub.min])/2 is introduced, where [[epsilon].sub.max] and [[epsilon].sub.min] are the maximum and minimum strain in one cycle, respectively. The ratcheting strain rate [[epsilon].sub.r], defined as the increment of ratcheting strain [[epsilon].sub.r] for each cycle, can also be utilized to describe the evolution of the ratcheting.

The modified model was employed to simulate the tension-tension cyclic tests of PC under various loading levels with/ without peak holding time in this subsection. Firstly, the Anand model and the modified model were used to predict the ratcheting behavior of PC for a chosen cyclic loading case. Comparison of the simulated results from the two models and the experiment one demonstrates that the modified one can describe the ratcheting behavior better. Then, the modified model is adopted to study the effect of various loading conditions on ratcheting of PC.



Comparison of the Two Models on Ratcheting. Figure 5a is a typical experiment stress-strain curve of PC in the case of mean stress 30 MPa and stress amplitude 15 MPa at stress rate of 1 MPa/s. Although the stress level of whole cyclic loading process is below the macroyield point, the accumulation of ratcheting strain remains significant even after the unloading. From the hysteresis loops in the first 1-5th, 50th, 100th, and 200th cycles respectively, it can be summarized that (i) the loop of stress-strain curve and the ratcheting strain move along the direction of nonzero mean stress; (ii) the hysteresis loops become more and more narrow with the increasing number of cycles; (iii) the progressive accumulation rate of strain at valley is faster than that at peak (the details are illustrated in Fig. 5d).

Figure 5b shows the corresponding stress-strain curves simulated by the Anand model. While the load level of whole ratcheting is always below the macroyield point, i.e., the recoverable elastic part of deformation plays an import rule, the simulated result fairly mimics the movement of the hysteresis loops along the direction of mean stress even without the consideration of the influence of previous deformation on the following cycle. However, the individual loops of stress-strain curve such as in the 50th, 100th, and 200th cycle keep the same shape and size. This does not agree with the experimental findings. The peak and valley strain are also found increasing at the same rate predicted by the Anand model with details shown in Figure 5d. Despite of its success in monotonic loading cases, the Anand model should be enhanced accordingly before its direct employment for modeling the ratcheting behavior of PC.


The simulated results by the modified model in the case of 30 [+ or -]15 MPa tension-tension cyclic loading are illustrated in Figure 5c. The width of the simulated hysteresis loops decrease with the increasing number of cycles, as well as the valley strain evolves faster than the peak one. This agrees well with the experiment results.

Although the valley strain of the modified model is still larger than that of experiment as shown in Figure 5d, the overall characteristics of ratcheting behavior of PC can be reproduced quit well using the modified model. In the following subsection, we will only focus on the modified model.


Simulation of Ratcheting via the Modified Model. To evaluate the influences of mean stress, stress amplitude, stress rate, and peak holding time respectively, Figures 6-9 summarize the experimental and corresponding simulated ratcheting deformation of PC under tension-tension cyclic loading with different loading conditions.

Figure 6 illustrates the ratcheting strain versus the number of cycles with different mean stresses (i.e., with a stress amplitude of 10 MPa and various mean stresses of 20, 30, 40 MPa, respectively) at the loading rate of 1 MPa/s. Figure 7 shows the ratcheting strain versus the number of cycles with different stress amplitudes (i.e., with a mean stress of 10 MPa and various stress amplitudes of 20, 30, 40 MPa respectively) at the loading rate of 1 MPa/s. It can be observed that the modified model provides a reasonably good prediction of the correlation between the ratcheting strain and the mean stress and stress amplitude. From Figures 6 and 7, one can also found that the modified model reproduces the phenomenon that the mean stress has more significant effect on the ratcheting strain of PC than the stress amplitude does. Different from the experimental results in which ratcheting strain tends to be saturated, the ratcheting strain rate from simulation does not approach zero even after certain cycles. Clearly, in such case, the peak stress reaches the 50 MPa (e.g., for the cases of 40 [+ or -] 10 and 30 [+ or -] 20 MPa at 1 MPa/s). It may be due to the fact that, the peak stress is close to the magnitude level of the yield strength of PC, and the viscoplastic deformation mechanism would contribute more for simulation, resulting in more plastic flow accumulation during the cyclic loading.


The rate-dependence behavior is also important for modeling cyclic behavior of polymers. Figure 8 shows the effect of stress rate, 1 and 10 Mpa/s, respectively, on ratcheting behavior of PC for the loading case of 40 [+ or -] 10 MPa. It demonstrates that the dependence of ratcheting on the stress rate can be reasonably predicted using the modified model. The ratcheting strain at a higher stress rate is smaller than that at a lower one. The ratcheting strain rate shows the similar tendency too, because a smaller stress rate represents a longer loading time per cycles which may lead to more viscous strain.

Figure 9 exhibits the ratcheting strain vs. the number of cycles with different peak holding times, 0, 10, and 60 s respectively, for the loading case of 30 [+ or -] 10 MPa. The modified model also can predict the influence of peak holding time on the ratcheting behavior of PC. Presenting a better prediction for longer holding time (comparing the cases of 10 and 60 s peak holding time with the case of 0 s), the modified model can describe the successive strain accumulation during the tension--tension ratcheting.



While the modified model shows the capability to reasonably describe the uniaxial tension-tension ratcheting of PC, further modeling work is undergoing for the multi-axial ratcheting, as well as the temperature effect on the ratcheting.


In this paper, a modified viscoelastic-plastic constitutive model is proposed to describe the stress-controlled tension-tension ratcheting behaviors of PC.

1. The evolution equations of deformation resistance for the viscoelastic micromechanism are introduced. Thus the influence of the accumulated viscoresistance on ratcheting deformation can be considered;

2. The modified model can reproduce the shape evolution of stress-strain hysteresis loops with the number of cycles and is in good agreement with the experimental results;

3. The modified model also shows good capability to predict the influences of mean stress, stress amplitude, loading rate, and peak holding time on the ratcheting behaviors of PC.

For a stress-controlled ratcheting test, the viscous and plastic deformation exists and accumulates simultaneously. Due to the inherent complexities of ratcheting behavior and polymeric materials, further exploration on the accurate description of polymers ratcheting behavior is essential and of significant value.


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Cheng Kai Jiang, Han Jiang, Jian Wei Zhang, Guo Zheng Kang

Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China

Correspondence to'. H. Jiang; e-mail: Contract grant sponsor: National Natural Science Foundation of China; contract grant numbers: 11172249 and 11272269; contract grant sponsor: Program for New Century Excellent Talents in University (NCET-12-0938).

DOI 10.1002/pen.24148

Published online in Wiley Online Library (
TABLE 1. Material parameters of PC (the Anand model).

Basic physical parameters
  Young's modulus E                                  2.07GPa
  Pressure sensitive parameter                        0.138
  Reference strain rate [[upsilon].sub.0]       0.0006 [s.sup.-1]
Viscoplasticity parameters ([alpha] = 0)
  The deformation resistance                     35 MPa/38.2 MPa
    (initial/saturation value)
  The free volume (initial/saturation                0/0.001
    value) [[phi].sub.i]/[[phi]]
  Strain rate sensitivity parameter                   0.027
  [h.sub.0]                                           6.7GPa
  [g.sub.0]                                           0.014
  h                                                    559
Visco-elasticity parameters ([alpha] =
  The deformation resistance                   7 MPa/20 MPa/30 MPa
  The backstress moduli (initial value)     6000 MPa/6000 MPa/6000 MPa
  The backstress moduli (final value)        400 MPa/280 MPa/180 MPa
Strain rate sensitivity parameter               0.027/0.027/0.027
  [c.sup.([alpha])]                          3.5 TPa/1.6 TPa/1.5 TPa

Note: the value of ap from Spitzig and Richmond [42].

TABLE 2. Additional material parameters of PC (the modified model
in Eq. 7).

Material parameters in Eq. 7 ([alpha] = l,2,3)
  [s.sup.([alpha]).sub.i]                         7 MPa/20 MPa/30 MPa
  [s.sup.([alpha]).sub.sat]                       40 MPa/40 MPa/40 MPa
  [d.sup.([alpha])]                               75 MPa/60 MPa/50 MPa
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Author:Jiang, Cheng Kai; Jiang, Han; Zhang, Jian Wei; Kang, Guo Zheng
Publication:Polymer Engineering and Science
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Geographic Code:1USA
Date:Nov 1, 2015
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