Abstract
Abstract Observations are reported on isotactic polypropylene/organically modified nanoclay hybrids with concentrations of filler ranging from 0 to 5 wt.% in cyclic tensile tests with a stress-controlled program (oscillations between various maximum stresses and the zero minimum stress). A pronounced effect of nanofiller is demonstrated: reinforcement with 2 wt.% of clay results in strong reduction of maximum and minimum strains per cycle and growth of number of cycles to failure compared with neat polypropylene. To rationalize these findings, a constitutive model is developed in cyclic viscoelasticity and viscoplasticity of polymer nanocomposites. Adjustable parameters in the stress-strain relations are found by fitting experimental data. The model correctly describes the growth of the ratcheting strain and shows that fatigue failure is driven by a pronounced increase in plastic strain in the crystalline phase. To assess the influence of loading conditions on the changes in the material parameters, experimental data on polypropylene are studied in cyclic tests with a strain-controlled program (oscillations between fixed maximum and minimum strains) and a mixed program (oscillations between various maximum strains and the zero minimum stress). Numerical simulation confirms the ability of the model to predict the evolution of stress-strain diagrams with the number of cycles.

1. To report observations in uniaxial tensile cyclic tests with a stress-controlled program on polypropylene/clay hybrids with various concentrations of filler and to demonstrate that the presence of nanoclay induces a strong decrease in maximum and minimum strains per cycle and growth of number of cycles to failure. 2. To develop constitutive equations in cyclic viscoelasticity and viscoplasticity of polymer nanocomposites and to find adjustable parameters in the stress-strain relations by fitting the observations. 3. To demonstrate ability of the model to predict the mechanical response in cyclic tests with different deformation programs (stress-controlled, strain-controlled, and mixed) and to examine the effect of loading conditions on evolution of materials parameters.

A number of studies dealt in the past five years with the viscoelastoplastic behavior of polymers and polymer composites subjected to cyclic loading (Drozdov and Christiansen 2007; Yakimets et al. 2007; Sullivan 2008; Drozdov 2009; Mizuno and Sanomura 2009; Ayoub et al. 2010; Bouchart et al. 2010) . The stress-strain relations developed in those works were confined, however, to the description of the first cycle of loading-unloading and could not be applied directly to fitting observations in multi-cycle tests. The first attempts to predict the response of polymers subjected to severe (hundreds of cycles) cyclic deformations were undertaken by Ayoub et al. (2011) and Drozdov (2011) .

Constitutive models in viscoplasticity of crystalline materials (where the viscoelastic phenomena are of secondary importance) under cyclic loading with stress-controlled programs were critically evaluated by Bari and Hassan (2002) , Chaboche (2008) , Kang (2008) , and Sai (2011) . It was pointed out that the governing equations captured some characteristic features of observations on metals, but (i) their ability to predict evolution of ratcheting strain was far from being satisfactory, and (ii) they failed to predict the mechanical response under loading with one program when material parameters are found by fitting stress-strain diagrams under deformation with another program.

To develop a model for the mechanical behavior polymer/clay nanocomposites that involves a reasonable number of adjustable parameters, a two-step approach is proposed. At the first step, stress-strain relations are derived for an individual cycle of loading-retraction. These relations contain several material parameters that are treated as constants. At the other step, these parameters are allowed to change slowly with number of cycles, and kinetic equations for their evolution are introduced. Coefficients in the kinetic equations are found by fitting observations along the first several cycles of loading-retraction. Validity of these relations is confirmed by comparing predictions of the model with observations in cyclic tests where number of cycles strongly exceeds that used for determination of material constants.

The exposition is organized as follows. Observations in uniaxial tensile cyclic tests with stress-controlled deformation program are reported in Sect. 2. Stress-strain relations for the viscoelastic and viscoplastic responses of polymer nanocomposites under tensile cyclic deformation are presented in Sect. 3 (derivation of constitutive equations for an arbitrary threedimensional deformation with small strains is given in the Appendix). Adjustable parameters in the governing equations are determined in Sect. 4 by fitting the experimental data. Experimental investigation and modeling of cyclic deformations of polypropylene with mixed and strain-controlled programs is performed in Sects. 5 and 6, respectively. Concluding remarks are formulated in Sect. 7.
Experimental results
Isotactic polypropylene Moplen HP 400R (density 0.90 g/cm 3 , melt flow rate 25 g/10 min, melting temperature 161
Experimental results
• C) was purchased from Albis Plastic Scandinavia AB (Sweden). Maleic anhydride grafted polypropylene Eastman G 3015 (acid number 15 mg KOH/g) was supplied by Eastman Chemical Company (USA). Organically modified montmorillonite nanoclay Delitte 67G was donated by Laviosa Chimica Mineraria S.p.A. (Italy).
Experimental results
Hybrid nanocomposites were manufactured in a two-step process ). At the first step, a masterbatch was prepared in a twin-screw extruder Brabender PL2000 with processing temperature 200
Experimental results
• C, screw speed 200 rpm, and throughput 5 kg/h. The masterbatch contained 40 wt.% of polypropylene, 40 wt.% of maleic anhydride grafted polypropylene, and 20 wt.% of nanoclay (clay/compatibilizer proportion 1:2). Prior to extrusion, all components were dried in an oven at 80
Experimental results
• C overnight. At the other step, pellets of the masterbatch were mixed with polypropylene in various proportions corresponding to nanoclay concentrations χ = 0, 2, and 5 wt.%. Dumbbell specimens for tensile tests (ASTM standard D638) with cross-sectional area 10.2 mm×4.2 mm were molded by using injection-molding machine Arburg 320C.
Experimental results
Mechanical tests were performed by means of universal testing machine Instron-5569 equipped with an electro-mechanical sensor for control of longitudinal strains. The tensile force was measured by a 5 kN load cell. The engineering stress σ was determined as the ratio of axial force to cross-sectional area of undeformed specimens.
Experimental results
Two series of uniaxial tensile tests were conducted at room temperature. The first series involved five ratcheting tests with various maximum stresses σ max , minimum stress σ min = 1 MPa, and cross-head speedḋ = 100 mm/min (that corresponded to strain rate˙ = 1.7 · 10 −2 s −1 ). A specimen was stretched with cross-head speedḋ up to maximum stress σ max , retracted down to minimum stress σ min with the same cross-head speed, reloaded up to maximum stress σ max with cross-head speedḋ, unloaded down to minimum stress σ min with the same cross-head speed, etc. The tests involved 200 cycles of loading-retraction (some of them were terminated earlier due to breakage of specimens). Each test was carried out on a new sample and repeated by twice to confirm reproducibility of measurements.
Experimental results
The minimum stress σ min = 1 MPa was chosen instead of σ min = 0 to avoid buckling of samples. The cross-head speedḋ = 100 mm/min was selected to ensure that maximum duration of cyclic tests did not exceed 20 min (the conventional duration of short-term creep and relaxation tests).
Experimental results
Ratcheting tests on neat polypropylene (χ = 0) were performed with maximum strains polymer/clay hybrid ), which demonstrated that nanocomposites with clay content of 1 to 2 wt.% had the highest resistance to creep flow.
Experimental results
The fact that some critical concentration χ cr exists (such that enhancement of mechanical properties of nanohybrids is observed at χ < χ cr only) has been reported by several authors (Hsieh et al. 2004; Bikiaris et al. 2006) . It is explained by (i) formation of a stable filler network structure in polymer nanocomposites (Wang et al. 2006; Su et al. 2011) , (ii) development of constrained regions where mobility of chains is severely restricted by surrounding clay platelets and their stacks (Zhang and Loo 2009) , and (iii) overlapping of interfacial layers of neighboring nanoparticles (Mortezaei et al. 2011) .
Experimental results
In the other series of experiments, ratcheting tests were conducted with cross-head 100 mm/min, minimum stress σ min = 1 MPa, and various maximum stresses σ max up to breakage of specimens. This series involved eight tests on neat polypropylene, three tests on nanocomposite with χ = 2 wt.%, and 5 tests on polymer/clay hybrid with χ = 5 wt.%. Observations are depicted in Fig. 4 where maximum stress σ max is plotted versus number of cycles to failure n f . Following common practice, the semi-logarithmic plot is used with log = log 10 . For each nanoclay content χ , experimental data are approximated by the Eyring formula
Experimental results
where σ 0 and σ 1 are calculated by the least-squares method. Fig. 4 demonstrates that 1. Equation (1) adequately describes the data, while σ 1 is weakly affected by concentration of nanoclay. 2. Reinforcement of polypropylene causes strong (by two orders of magnitude) growth of number of cycles to failure. 3. The most pronounced enhancement of fatigue resistance is reached when clay content equals 2 wt.%. Based on these observations, we confine ourselves to study of the mechanical response of nanocomposites with χ = 0 and 2 wt.% only. Experimental stress-strain diagrams for these nanohybrids under cyclic loading with the highest maximum stresses (σ max = 32 MPa at χ = 0 wt.% and σ max = 35 MPa at χ = 2 wt.%) are depicted in Figs. 5 and 6, and corresponding dependencies of maximum and minimum strains on n are plotted in Figs. 7 and 8. The data are chosen for the detailed analysis as the number of cycles to failure in ratcheting tests practically coincide. The following conclusions are drawn from Figs. 5, 6, 7 and 8:
Experimental results
1. The stress-strain curves under loading and unloading are strongly nonlinear.
Experimental results
2. An increase in number of cycles n results in growth of hysteresis energy (calculated as the area between subsequent loading and unloading paths) and clockwise rotation of stress-strain diagrams. (9) 3. Reinforcement of polypropylene with nanoclay induces a strong decay in maximum strain per cycle: after 46 cycles of oscillations, max equals 0.158 at χ = 0 wt.%, σ max = 32 MPa versus 0.096 at χ = 2 wt.%, σ max = 35 MPa and 0.040 at χ = 2 wt.%, σ max = 32 MPa. 4. Although fatigue failure of specimens with χ = 0 and 2 wt.% occurs after the same number of cycles (n f = 48), maximum max and minimum min strains in ratcheting tests on nanocomposite with χ = 2 wt.% are noticeably lower than those measured on neat polypropylene.
Experimental results
Experimental data reveal good reproducibility of measurements: maximum deviation between strains measured on different specimens under the same loading conditions does not exceed 6%.
Constitutive model
To rationalize these observations, constitutive equations are derived for the viscoelastic and viscoplastic responses of nanocomposites under cyclic deformation. A two-step approach is employed for modeling. At the first step, stress-strain relations are developed in cyclic viscoelastoplasticity. These relations involve several adjustable parameters that are treated as constants for an individual cycle of loading-unloading. At the other step, kinetic equations are suggested for evolution of these quantities with number of cycles. Coefficients in the kinetic equations are found by fitting experimental data along the first several cycles (30 to 50) of loading-retraction. Validation of these equations is performed by comparison of the model predictions with observations in cyclic tests with relatively large numbers of cycles (that strongly exceed the number of cycles used to determine material constants).
Stress-strain relations
Constitutive equations in viscoelasticity and viscoplasticity of polymer nanocomposites under an arbitrary three-dimensional deformation with small strains are developed in the Appendix. We present a simplified version of these relations for uniaxial tensile cyclic loading with constant strain rate˙ .
Stress-strain relations
Strain equals the sum of elastic e and plastic p strains
Stress-strain relations
The plastic strain p is split into the sum of two components
Stress-strain relations
that reflect plastic deformation in the crystalline and amorphous phases, respectively. The strain rate for plastic deformation in the crystalline phase is proportional to that for macro-
Stress-strain relations
The coefficient φ obeys the differential equations
Stress-strain relations
where a 1 and a 2 are non-negative coefficients. The strain rate for plastic deformation in the amorphous matrix is governed by the equations
Stress-strain relations
where R 1 , R 2 , S 1 , and S 2 are non-negative parameters. The function f (v) is determined by (A.7). The function Z(t, v) obeys the differential equation
Stress-strain relations
where Γ (v) is given by (A.5). The stress σ reads
Stress-strain relations
where E stands for Young's modulus. Stress-strain relations (2)-(8) involve nine adjustable parameters:
Stress-strain relations
The constants γ and Σ are found by fitting observations in short-term relaxation tests. Approximation of the relaxation curves reported in Drozdov et al. (2009) implies that these quantities are independent of χ and read γ = 0.11 s −1 , Σ = 13.1. Treatment of γ and Σ as constants means that the effect of cyclic viscoplasticity on the viscoelastic response is disregarded.
Stress-strain relations
The elastic modulus E is determined by matching stress-strain diagrams under stretching of virgin specimens. Approximation of the loading curves with cross-head speeḋ d = 100 mm/min reported in Drozdov et al. (2009) implies that E = 2.26 GPa at χ = 0 wt.% and E = 2.85 GPa at χ = 2 wt.%. The assumption that E is constant distinguishes the present approach from conventional concepts in damage mechanics that postulate a decay in elastic modulus under deformation.
Evolution equations
To reduce the number of adjustable parameters, it is postulated that R 1 is independent of n and reads R 1 = 1.
Evolution equations
To describe evolution of the remaining parameters, a 1 , a 2 , R 2 , S 1 , S 2 , with number of cycles, we introduce plastic work
Evolution equations
as the measure of changes in internal structure of nanocomposites driven by damage accumulation. The following scenario is proposed:
Evolution equations
1. Coefficients a 1 and a 2 decrease monotonically with number of cycles and vanish when n becomes relatively large. The decay in these parameters is described by the equations
Evolution equations
where a 10 , a 11 , a 20 , a 21 are constants, and W min p , W max p are plastic works at instants when strain reaches its minimum and maximum values, respectively. 2. Coefficient R 2 decreases monotonically with number of cycles and vanishes at relatively large values of n. Depending on the loading program, the decay in R 2 is described either by the logarithmic equation
Evolution equations
or by the linear equation
Evolution equations
where R 20 , R 21 are constants. 3. During an initial transition period (when a 1 and a 2 decay to zero), coefficients S 1 and S 2 strongly decrease with number of cycles. Afterwards, these quantities evolve linearly with plastic work
Evolution equations
where S 10 , S 11 , S 20 , S 21 are constants.
Evolution equations
After an initial transition period (along which parameters a 1 , a 2 , and R 2 vanish), mechanical response of a polymer/clay nanohybrid under cyclic deformation is determined by five adjustable parameters: (i) E stands for elastic modulus, (ii) γ determines relaxation rate, (iii) Σ characterizes distribution of relaxation times, (iv) S 1 and S 2 denote rates of plastic flow in the amorphous matrix under loading and unloading.
Fitting of observations
Adjustable parameters in the constitutive equations are found by fitting the experimental data depicted in Figs. 5 and 6. Each set of observations is matched separately.
Loading
The response of a polymer nanocomposite under stretching is determined by two parameters, a 1 and S 1 . These quantities are found by fitting observations with the help of the following algorithm. To approximate the first loading path, we fix some intervals [0,
Loading
, where a 1 and S 1 are located, and divide these intervals into J = 10 sub-intervals by the points (2)- (8) are integrated numerically from = 0 to = max , where max stands for maximum strain under stretching. Integration over time is performed by the Runge-Kutta method with step t = 0.01 s. Integrals in (6) and (8) Subsequent loading paths are approximated by means of the same algorithm. The only difference is that the right boundary of the interval [0, a • ] at the nth cycle is chosen from the condition a
Loading
• ≤ a 1 (n − 1), where a 1 (n − 1) is the best-fit value of a 1 at the (n − 1)th cycle. When a 1 vanishes at n th cycle, fitting of observations at the nth cycle with n > n is performed with the only adjustable parameter S 1 .
Unloading
The response of a polymer nanocomposite under retraction is determined by three parameters, a 2 , R 2 , and S 2 . These quantities are found by means of the above algorithm where the right boundaries of all intervals [0,
Unloading
, and S • ≤ S 2 (n − 1), which guarantee monotonic decay of these coefficients with number of cycles. After the initial transition period, when a 2 and R 2 vanish, this restriction on S 2 is removed.
Discussion
When a 1 , a 2 , R 2 , S 1 , S 2 are determined for each cycle of loading-retraction, plastic work W p is calculated by integration of the stress-strain relations, and the adjustable parameters are plotted versus W p in Figs. 9-13. (9) where coefficients are calculated by the leastsquares technique. These figures demonstrate that a 1 and a 2 decrease linearly with plastic work and vanish when fatigue failure of specimens occurs. The best-fit values of a 1 and a 2 at the first and last cycles are weakly affected by nanoclay content χ . However, slopes a 11 and a 21 for nanocomposite with χ = 2 wt.% of filler exceed those for neat polypropylene practically by twice (due to more pronounced growth of W p at χ = 0 wt.%).
Discussion
The experimental diagrams R 2 (W max p ) are presented in Fig. 11 where the data are approximated by (10). This figure shows that the decay in R 2 with plastic work is weakly affected by clay content (R 21 at χ = 2 wt.% exceeds that at χ = 0 wt.% by 17%).
Discussion
Evolution of coefficients S 1 and S 2 with plastic work W p is illustrated in Figs. 12 and 13. According to these figures, S 1 and S 2 decrease monotonically and become independent of W p at the end of the initial transition period. The ultimate value of S 1 is weakly affected by (12) clay content (S 10 at χ = 2 wt.% exceeds S 10 at χ = 0 wt.% by 7%), whereas the ultimate value of S 2 at χ = 2 wt.% exceeds that for neat polypropylene by 35%.
Discussion
Comparison of coefficients in the stress-strain relations leads to the conclusion that a strong increase in fatigue resistance due to reinforcement of polypropylene with nanoclay may be associated with growth of rates of plastic flow in the amorphous matrix under loading S 1 and retraction S 2 .
Numerical simulation
Although the above analysis seems reasonable, a number of questions arises:
Numerical simulation
1. Adjustable parameters in the constitutive equations were found to ensure the best fit of the experimental stress-strain diagrams. How accurate approximation of observations can be reached within the proposed model? To provide some answers to these questions, numerical simulation is conducted of the stress-strain relations with the adjustable parameters depicted in Figs. 9-13. To examine the quality of the approximations (9)- (12), integration of the constitutive equations for the first 25 cycles is performed with the best-fit parameters a 1 , a 2 , R 2 , S 1 , S 2 , while integration for subsequent cycles of loading-retraction is carried out by using (9), (10), and (12). Results of calculations are depicted in Figs. 5 and 6. These figures demonstrate excellent agreement between the experimental stress-strain diagrams and predictions of the model for the first 40 cycles (stages of primary and secondary fatigue) and quite an acceptable prediction of observations at the stage of tertiary fatigue (slight deviations between experimental data and results of simulation are visible along the last two cycles before breakage only).
Numerical simulation
To examine validity of the model, we compare experimental dependencies max (n) and min (n) with results of numerical analysis for 50 cycles of loading-retraction in Figs. 7 and 8. These figures reveal good agreement between the observations and the results of numerical analysis up to breakage of specimens under ratcheting.
Numerical simulation
To understand why failure of specimens is observed under stress-controlled loading, maximum both concentrations of filler (appropriate data are not presented as the model is directed to describe the mechanical response at small strain only).
Mixed deformation program
To demonstrate ability of the constitutive model to describe the viscoelastic and viscoplastic responses under cyclic deformation with other loading programs, the mechanical behavior of polypropylene is analyzed in uniaxial tensile tests with a mixed program, when a specimen is stretched up to some maximum strain max and unloaded down to a fixed minimum stress σ min . A series of uniaxial cyclic tests is performed at room temperature with maximum strains max = 0.08, 0.12, and 0.16, minimum stress σ min = 1 MPa, and cross-head speedḋ = 100 mm/min. Each test involved 150 cycles of loading-retraction.
Mixed deformation program
A typical stress-strain diagram (corresponding to max = 0.12) is reported in Fig. 16 . This figure shows that in cyclic tests with mixed program:
Mixed deformation program
1. Loading and unloading paths are strongly nonlinear. 2. Hysteresis energy decreases noticeably with number of cycles. 3. The stress-strain diagram rotates counter-clockwise with growth of n.
Mixed deformation program
Evolution of maximum stress per cycle σ max and minimum strain per cycle min with n is illustrated in Fig. 17 which shows that σ max substantially decreases within the first 20 cycles and becomes practically constant afterwards, whereas min increases monotonically with number of cycles.
Mixed deformation program
Adjustable parameters in the constitutive equations are determined by matching the stress-strain curve for the first 50 cycles of oscillations. These quantities are presented in Figs. 18 and 19 , where the data are approximated by (9), (11), and (12). The fitting procedure coincides with that used in the analysis of cyclic tests with stress-controlled program. (11) is not dramatic: an accurate approximation of the data depicted in Fig. 18 shows that (10) adequately fits the experimental dependence for the first three-four cycles, whereas (11) is more suitable for subsequent cycles of loading-retraction.
Mixed deformation program
According to Fig. 19 , parameter S 1 increases weakly, but monotonically with plastic work. After the transition period, its dependence on W min p is correctly approximated by (12). Coefficient S 2 decreases with W max p within the transition period and grows with plastic work afterwards.
Mixed deformation program
To validate (12), numerical simulation is conducted of the stress-strain relations for 150 cycles of loading-retraction. For the first 50 cycles of oscillations, integration of the constitutive equations is performed with the adjustable parameters depicted in Figs. 18 and 19, whereas for the other 100 cycles, (12) is used. Results of numerical analysis are reported in To examine a difference between the viscoelastoplastic responses of polypropylene in stress-controlled cyclic tests (where fatigue failure occurs) and cyclic tests with a mixed program (where no breakage of samples is observed), evolution of maximum and minimum plastic strains with number of cycles is studied numerically. Results of numerical simulation are depicted in Fig. 20 . This figure shows that maximum plastic strains The data depicted in Figs. 19 and 20 unambiguously demonstrate that plastic strains p , p1 , and p2 cannot be used as internal variables: Fig. 19 show that S 2 strongly increases with number of cycles, while maximum plastic strains remain independent of n.
Mixed deformation program
To check whether W p is a proper candidate for internal variable, stress-strain diagrams in cyclic tests with mixed loading program and various max are approximated until coefficients a 1 , a 2 and R 2 vanish. To compare rates of their decay with W p , auxiliary variables are introducedā
Mixed deformation program
where a 2 (1) and R 2 (1) stand for coefficients a 2 and R 2 at first retraction (these parameters read 3.0 and 0.65 at max = 0.08, 6.1 and 0.31 at max = 0.12, 8.0 and 0.22 at max = 0.16). The data are depicted in Fig. 21 (9) and (11), (13) with coefficients calculated by the least-squares method. Although scatter of the data is noticeable, coincidence of the slopes of corresponding curves in Fig. 21 confirms that W p may be treated as an internal variable that accounts for damage accumulation under cyclic deformations with mixed program.
Strain-controlled deformation program
Our aim now is to demonstrate ability of the constitutive model to predict observations under cyclic deformation with a strain-controlled program (oscillations between fixed maximum strain max and minimum strain min ). For this purpose, uniaxial tensile cyclic test (100 cycles) was performed on polypropylene with cross-head speedḋ = 100 mm/min, maximum strain max = 0.07, and minimum strain min = 0.04. The test was conducted on the same grade of polypropylene as cyclic tests with stress-controlled and mixed programs. The only difference is that dumbbell specimens were injection-molded under different conditions (a lower temperature of the mold) which led to lower degree of crystallinity and lower Young's modulus (E = 1.42 GPa).
Strain-controlled deformation program
The experimental stress-strain diagram is reported in Fig. 22 . To avoid overlapping of observations, only data for the first, 10th, and 30th cycle are depicted. The following conclusions are drawn:
Strain-controlled deformation program
1. Loading and unloading paths of the stress-strain diagram are strongly nonlinear. 2. Hysteresis energy decreases noticeably with number of cycles. 3. Stress-strain curves rotate counter-clockwise with growth of n. 4. The secant elastic modulus E s (defined as slope of the straight line connecting points when strain rate changes its sign) increases monotonically with n (cyclic strengthening). (9) and (10) Maximum σ max and minimum σ min stresses are plotted versus number of cycles in Fig. 23 . This figure shows that both stresses decrease strongly with n, but the decay in σ min occurs more pronouncedly. To find adjustable parameters in the constitutive equations, experimental stress-strain diagrams are approximated for the first 30 cycles of oscillations. Coefficients a 1 , a 2 , and R 2 are plotted versus plastic work W p in Fig. 24 where the data are approximated by (9) and (10). This figure shows that (9) and (10) correctly fit the observations.
Strain-controlled deformation program
Comparison of Figs. 18 and 24 demonstrates the same kinetics of evolution of coefficients a 1 and a 2 (these quantities decay to zero practically linearly with W p ), but different kinetics of decrease in R 2 . Under cyclic deformation with mixed program, the latter quantity vanishes simultaneously with a 1 and a 2 and decreases linearly with W p , whereas under cyclic deformation with strain-controlled program, R 2 weakly decays with W p following the same pattern (10) as under stress-controlled cyclic loading. Parameters S 1 and S 2 are plotted versus plastic work W p in Fig. 25 . The data after the initial transition period (within which a 1 and a 2 vanish) are matched by (12) with coefficients determined by the least-squares method. Figure 25 confirms accuracy of approximation of the data by (12).
Strain-controlled deformation program
Comparison of Figs. 12, 13, 19, and 25 reveals different kinetics of evolution of rates of plastic flow in the amorphous phase under cyclic loading: S 1 and S 2 (i) remain constant under stress-controlled program, (ii) increase under mixed program, and (iii) decrease under strain-controlled program. This observation can be taken into account with the help of the following generalization of (12):
Strain-controlled deformation program
where H stands for the Heaviside function:
Strain-controlled deformation program
To confirm validity of approximations (9), (10), and (12), numerical simulation is conducted of the stress-strain relations for 100 cycles of loading-retraction. For the first 30 cycles, integration of the governing equations is performed with parameters a 1 , a 2 , R 2 , S 1 , S 2 depicted in Figs. 24 and 25, whereas for subsequent cycles, (9), (10), and (12) are employed. Results of numerical analysis are depicted in Fig. 23 . This figure reveals good agreement between the experimental data for maximum and minimum stresses and predictions of the model.
Strain-controlled deformation program
To characterize plastic flow in polypropylene under strain-controlled deformation program, maximum and minimum plastic strains are plotted versus number of cycles in Fig. 26 . This figure demonstrates that plastic deformation in the crystalline phase remains constant and shows slow growth of total plastic strains 
Concluding remarks
Observations have been reported on polypropylene/nanoclay hybrids with various concentrations of filler in tensile cyclic tests with stress-controlled program. Experimental data show that reinforcement of polypropylene with nanoclay resulted in strong (by several times) reduction in ratcheting strain and substantial (by two orders of magnitude) growth of number of cycles to failure. The most pronounced improvement of fatigue resistance is reached when concentration of nanoclay equals 2 wt.%. A constitutive model is developed in cyclic viscoelasticity and viscoplasticity of polymer nanocomposites by using a two-step approach. First, stress-strain relations are derived for an individual cycle of loading-retraction by using the Clausius-Duhem inequality. Afterwards, some coefficients in the governing equations are presumed to slowly change with number of cycles, and kinetic equations are proposed for evolution of these quantities driven by damage accumulation. Specific plastic work is selected as an internal variable in evolution equations for rates of viscoplastic flow, and this choice is confirmed by comparison of observations in cyclic tests with mixed deformation program and various maximum strains.
Concluding remarks
The stress-strain equations involve eight material parameters. Three of them are determined in independent tests (tension with a constant strain rate and short-term relaxation tests). The other five quantities are split into two groups: three parameters strongly decrease and vanish within an initial transition period (a couple of dozens of cycles); the last two parameters slowly evolve with the number of cycles following (14) . Ability of the model to describe experimental data is examined by analyzing observations in tensile tests with stress-controlled (oscillations between fixed minimum and maximum stresses), mixed (oscillations between a maximum strain and a minimum stress), and strain-controlled (oscillations between fixed minimum and maximum strains) programs. Coefficients in the constitutive equations are found by fitting stress-strain diagrams along the first several cycles (30 to 50, depending on the deformation program). Afterwards, predictions of the model are compared with observations in tests where number of cycles strongly exceeds that used for determination of material constants. Good agreement is revealed between results of numerical simulation and experimental data.
Concluding remarks
Comparison of the viscoelastic and viscoplastic responses of polymer specimens subjected to different deformation programs reveals the presence of two independent mecha-nisms of damage accumulation responsible for different kinetics of evolution of coefficients a 1 and a 2 , on the one hand, and R 2 , on the other. The linear decay in a 1 and a 2 with plastic work W p observed under all deformation programs may be attributed to lamellar fragmentation in spherulites. The monotonic reduction in R 2 whose kinetics coincides with that for a 1 and a 2 under mixed deformation program and differs noticeably from the kinetics of decrease in a 1 and a 2 under stress-and strain-controlled loading programs may be associated with growth of micro-voids in the amorphous matrix. Experimental data reported in this study are, however, insufficient for an unambiguous explanation of this phenomenon and further analysis is required.
Concluding remarks
Numerical simulation demonstrates that fatigue failure of specimens under stresscontrolled loading is induced by a pronounced growth of plastic deformation in the crystalline phase. In cyclic tests with mixed and strain-controlled programs (where breakage of specimens does not occur) maximum and minimum plastic strains in crystallites remain practically independent of number of cycles.
Concluding remarks
Comparison of material parameters for neat and clay-reinforced polypropylene leads to the conclusion that enhancement of fatigue resistance may be ascribed to acceleration of plastic flow in the matrix driven by sliding of clay platelets in their stacks, as well as sliding of polymer chains along the surfaces of clay particles.
Appendix: Constitutive model
This section focuses on derivation of constitutive equations in viscoelastiplasticity of polymer nanocomposites for an individual cycle of loading-retraction. Our purpose is to develop stress-strain relations with a relatively small number of adjustable parameters to be applied for description of the mechanical response under multi-cycle loading with an arbitrary threedimensional deformation at small strains.
Appendix: Constitutive model
To construct a model with a reasonable number of material constants, a homogenization concept is employed. A nanocomposite with a complicated microstructure (a semicrystalline matrix reinforced with randomly distributed clay platelets and their stacks) is replaced with an equivalent isotropic medium, whose mechanical response resembles that of the hybrid. An incompressible, inhomogeneous, transient, non-affine network of chains bridged by junctions is chosen as the equivalent continuum (Drozdov and Christiansen 2007) .
Appendix: Constitutive model
The incompressibility assumption means that deformation of nanocomposite is volumepreserving. This hypothesis is confirmed by our observations in uniaxial tensile tests with various cross-head speeds where longitudinal and transverse strains were measured simultaneously .
Appendix: Constitutive model
The concept of transient networks was suggested for the analysis of time-dependent behavior of polymers by Tanaka and Edwards (1992) . This theory associates relaxation of stresses with rearrangement of chains in a polymer network: detachment of active chains from their junctions and attachment of dangling chains to the network.
Appendix: Constitutive model
The assumption about heterogeneity of the network means that the polymer network is thought of as an ensemble of meso-regions where rearrangement of chains occurs with different rates. This statement is tantamount to the conventional assertion about the existence of a spectrum of relaxation times.
Appendix: Constitutive model
Non-affinity of the network means that junctions between chains slip with respect to their reference positions under deformation. Slippage of junctions reflects plastic flow in polymer nanocomposite. To account for different kinetics of plastic deformation in amorphous and crystalline domains, the entire plastic strain is split into the sum of two components whose evolution is governed by different flow rules.
Appendix: Constitutive model
A shortcoming of the present model is that it disregards interactions between different phases, in particular the presence of interfaces between the matrix and clusters of clay platelets (Sheng et al. 2004; Qiao and Brinson 2009; Kaushik et al. 2011; Li et al. 2011a Li et al. , 2011b , as well as the presence of inter-phases between amorphous matrix and spherulites and near the fold and stem surfaces (Rastogi et al. 2007 ). This neglect may be explained by our attempt to make the model as simple as possible for (i) explicit account for several phases in a polymer nanocomposite leads to a substantial increase in number of adjustable parameters, which makes difficult predictions of the mechanical response, and (ii) a detailed analysis of deformations at interfaces and in inter-phases where relatively large deformations are observed (Kaushik et al. 2011; Li et al. 2011a Li et al. , 2011b ) requires application of a more sophisticated apparatus of continuum mechanics with finite strains.
A.1 Kinematics of plastic deformations
To describe plastic flow in the equivalent medium at small strains, the strain tensor for macro-deformationˆ is presented as the sum of elastic strain tensorˆ e and plastic strain tensorˆ pˆ =ˆ e +ˆ p .
A.1 Kinematics of plastic deformations
(A.1)
A.1 Kinematics of plastic deformations
With reference to conventional phenomenological models with two plastic elements connected in series, the plastic strain tensorˆ p is split into the sum of two componentŝ
A.1 Kinematics of plastic deformations
It is presumed that tensorsˆ p1 andˆ p2 describe inelastic deformations in crystalline and amorphous phases, respectively. As no special phase is introduced for the analysis of deformation in stacks of clay platelets, the amorphous phase is associated with an ensemble of polymer chains in the rubbery state with randomly distributed nanoparticles. The strain rate for plastic deformation in the crystalline phase is proportional to the strain rate for macro-deformation,
A.1 Kinematics of plastic deformations
where the non-negative function φ (i) vanishes in the undeformed state (which means that no plastic deformation occurs at very small strains), (ii) monotonically grows under active loading and decreases under retraction (which reflects stress-induced acceleration of plastic flow), and (iii) reaches its ultimate value φ ∞ = 1 at large strains (when the rates of plastic deformation and macro-deformation coincide). Changes in φ with time are governed by the differential equation (A.4) where the signs "+" and "−" correspond to loading and retraction, respectively, anḋ
A.1 Kinematics of plastic deformations
stands for the equivalent strain rate for macro-deformation. The coefficient a adopts different values a 1 and a 2 under active deformation and unloading. For uniaxial tensile cyclic deformation, which is the subject of this work, loading and unloading processes are determined unambiguously. For an arbitrary three-dimensional deformation, they can be defined following the approaches proposed by Bari and Hassan (2002) and Xia et al. (2005) .
A.1 Kinematics of plastic deformations
A.2 Heterogeneity of the network An inhomogeneous transient polymer network is composed of meso-domains with various activation energies for rearrangement of chains. The rate of separation of active chains from their junctions in a meso-domain with activation energy u is governed by the Eyring equation
A.1 Kinematics of plastic deformations
, where γ is an attempt rate, k B denotes Boltzmann's constant, and T stands for the absolute temperature. Confining the analysis to isothermal processes and introducing dimensionless activation energy v = u/(k B T ), we present this equation in the form
A.1 Kinematics of plastic deformations
Denote by N concentration of active chains in the equivalent network (Tanaka and Edwards 1992) . The number of active chains n 0 (v) in meso-domains with activation energy v (per unit volume) reads
A.1 Kinematics of plastic deformations
where f (v) stands for a distribution function of meso-domains. With reference to the random energy model (Derrida 1980) , the quasi-Gaussian expression is adopted for this function
A.1 Kinematics of plastic deformations
An advantage of (A.7) is that it involves the only adjustable parameter Σ > 0. The pre-factor f 0 is determined from the normalization condition ∞ 0 f (v) dv = 1. Parameters N , γ , and Σ are assumed to be independent of mechanical factors.
A.3 Rearrangement of chains
Rearrangement of a temporary network is described by a function n (t, τ, v) which equals the number (per unit volume) of temporary chains at time t ≥ 0 that have returned into the active state before instant τ ≤ t and belong to a meso-domain with energy v. The number of active chains in meso-domains with energy v at time t reads n(t, t, v) = n 0 (v), (A.8) while the number of chains that were active at the initial instant t = 0 and have not separated from their junctions until time t is n(t, 0, v). (t, τ, v) , (A.10) which mean that the number of active chains separating from their junctions per unit time is proportional to the total number of active chains in an appropriate meso-domain. Integration of (A.10) with initial conditions (A.6), (A.8), and (A.9) (A.24) provide constitutive equations for the viscoelastoplastic behavior of polymer nanocomposites under cyclic deformation. These equations involve three material constants, μ, γ , Σ , and six adjustable functions, a 1 , a 2 , R 1 , R 2 , S 1 , S 2 , with the following physical meaning:
A.3 Rearrangement of chains
• μ stands for an elastic modulus of a polymer nanocomposite, • γ and Σ characterize its linear viscoelastic behavior, • a 1 and a 2 describe irreversible deformation in the crystalline phase under loading and unloading, • R 1 , S 1 and R 2 , S 2 characterize plastic flow in the amorphous matrix under active deformation and retraction.

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