nanoclay hybrids

Computational Materials Science 53 (2012) 396–408

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Cyclic viscoelastoplasticity of polypropylene/nanoclay hybrids A.D. Drozdov a,⇑, J.deC. Christiansen b a b

Department of Plastics Technology, Danish Technological Institute Gregersensvej 1, Taastrup 2630, Denmark Department of Mechanical and Manufacturing Engineering, Aalborg University Fibigerstraede 16, Aalborg 9220, Denmark

a r t i c l e

i n f o

Article history: Received 3 February 2011 Received in revised form 25 July 2011 Accepted 4 September 2011 Available online 22 October 2011 Keywords: Polymer nanocomposite Viscoelasticity Viscoplasticity Ratcheting Constitutive modeling

a b s t r a c t Observations are reported on isotactic polypropylene/organically modified nanoclay hybrids in tensile cyclic tests at room temperature (ratcheting between fixed maximum and minimum stresses). A pronounced effect of nanofiller is demonstrated: reinforcement of polypropylene with 1 wt.% of clay results in reduction of maximum and minimum strains per cycle by several times and growth of number of cycles to failure by an order of magnitude. To rationalize these findings, a constitutive model is developed in cyclic viscoelastoplasticity of polymer nanocomposites. Adjustable parameters in the stress–strain relations are found by fitting experimental data in relaxation tests and cyclic tests (16 cycles of loading–unloading). It is demonstrated that the model correctly predicts growth of maximum and minimum strains per cycle with number of cycles and can be applied to predict fatigue failure of nanocomposites. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction This paper is concerned with experimental investigation and constitutive modeling of the mechanical response of polymer/clay nanocomposites in uniaxial tensile cyclic tests with a stress-controlled program (ratcheting between a minimum stress rmin and a maximum stress rmax). A number of studies on the viscoelastic and viscoplastic behavior of polymer/nanoclay hybrids revealed noticeable improvement of their mechanical properties due to the presence of nanoparticles [1–3]. This enhancement remains, however, rather modest compared with what has been expected from the effect of reinforcement about a decade ago [4,5]. Several hypotheses have been suggested to explain this gap: (i) inhomogeneity in the distribution of nanoparticles driven by clustering of clay platelets and insufficient intercalation and exfoliation of stacks [6,7], (ii) the presence of inter-inclusion interactions reducing stiffness of nanocomposites [8] and resulting in formation of a hierarchical structure of filler particles [9], (iii) weak adhesion between polymer chains and clay particles with high aspect ratios [10] causing significant embrittlement of hybrids [4], (iv) development of interphase zones surrounding nanoparticles, where mobility of chains is substantially reduced compared with that in the bulk matrix [11,12], etc. Keeping in mind discrepancies between expected and real mechanical properties of organic/inorganic nanohybrids, two directions of research have been formulated: (i) to investigate ⇑ Corresponding author. Tel.: +45 72 20 31 42; fax: +45 72 20 31 12. E-mail address: [email protected] (A.D. Drozdov). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.09.004

in detail physical nature of the observed disagreement, and (ii) to focus on more sophisticated (compared with uniaxial tension) deformation programs, where the effect of nanofiller is significant for engineering applications. The latter approach was initiated in [13–17] by demonstrating that reinforcement of polymers with nanoparticles dramatically enhances their creep resistance. It was shown in [18,19] that not only creep compliance in conventional creep tests, but also ratcheting strain in cyclic tests with a stress-controlled program were strongly reduced due to the presence of SiO2 nanoparticles. To the best of our knowledge, no studies have focused on constitutive modeling of this phenomenon. Observations on the mechanical response of polymers and polymer composites in ratcheting tests were recently reported in [20–24], to mention a few. The objective of this study is threefold: 1. To report observations at room temperature in uniaxial tensile cyclic tests with a stress-controlled program on polypropylene/nanoclay hybrids with various concentrations of filler (ranging from 0 to 5 wt.%) and to demonstrate a strong decrease in maximum and minimum strains per cycle due to the presence of nanoparticles. 2. To develop a constitutive model in cyclic viscoelastoplasticity of polymer nanocomposites and to find adjustable parameters in the stress–strain relations by fitting observations in cyclic tests and relaxation tests. 3. To demonstrate by numerical simulation that the constitutive equations can be employed for evaluation of number of cycles to failure in ratcheting tests.

A.D. Drozdov, J.deC. Christiansen / Computational Materials Science 53 (2012) 396–408

In the past five years, a number of studies have dealt with experimental and theoretical analysis of the time-dependent response of polymers and polymer composites subjected to cyclic loading [25–32]. The stress–strain relations developed in those works were confined, however, to the description of experimental data along the first cycle (loading–unloading) of a cyclic deformation program and could not be directly applied to fitting observations in multi-cycle tests. Constitutive equations in viscoplasticity of materials (mainly, metals) under cyclic loading with stress-controlled programs were recently suggested in [33– 35]. Several models were critically evaluated in [36–39]. It was pointed out that (i) the governing equations captured some characteristic features of observations, but (ii) their ability to predict evolution of ratcheting strain was far from being satisfactory. The first attempt to predict the response of polymers subjected to severe (hundreds of cycles) deformations based on observations during the first few cycles was undertaken in [40], where periodic loadings with fixed maximum strains and minimum stresses were analyzed. In the present work, another approach is proposed to modeling the viscoelastic and viscoplastic behavior of polymer/clay nanocomposites. To reduce the number of adjustable parameters in the stress–strain relations, a homogenization concept is applied [26]. A nanohybrid with a complicated microstructure is replaced with an equivalent medium whose mechanical behavior resembles that of the composite. As an equivalent medium, a non-affine transient heterogeneous network of polymer chains is chosen. Two characteristic features of the equivalent network are to be mentioned: (i) the plastic strain tensor is split into the sum of two components that characterize irreversible deformations in the matrix and inclusions, and (ii) the strain energy density of the composite equals the sum of mechanical energies stored in individual chains and the energy of interaction between chains and stacks of clay platelets. Adjustable parameters in the stress–strain relations are determined by approximating observations during the first 16 cycles of loading–retraction. It is revealed in the fitting procedure that majority of coefficients in the evolution equations reach their ultimate values after a short transition period (a few cycles of deformation). After this transition period, only two parameters (reflecting interaction between chains and clay platelets) proceed to slowly evolve with number of cycles. Simple differential equations are derived to describe their decay with intensity of plastic strain. Coefficients in these equations are found by fitting experimental dependencies of maximum and minimum strains per cycle on number of cycles. The exposition is organized as follows. Observations on polypropylene/clay hybrids in uniaxial tensile cyclic tests and relaxation tests are reported in Section 2. Constitutive equations in cyclic viscoelasticity and viscoplasticity of polymer nanocomposites are developed in Section 3. Adjustable parameters in the stress–strain relations are determined in Section 4 by fitting the experimental data. A brief discussion of our findings is provided in Section 5. Concluding remarks are formulated in Section 6.

2. Experimental results Isotactic polypropylene Moplen HP 400R (density 0.90 g/cm3, melt flow rate 25 g/10 min, melting temperature 161 °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). Hybrid nanocomposites were manufactured in two steps following the procedure described in [41]. First, a masterbatch

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was extruded in a twin-screw extruder Brabender PL2000 with processing temperature 200 °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 °C overnight. Pellets of the masterbatch were mixed with polypropylene in various proportions corresponding to nanoclay concentrations v = 0, 1, 3, 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. 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 r was determined as the ratio of axial force to cross-sectional area of undeformed specimen. Two series of uniaxial tensile tests were conducted at room temperature. The first series aimed to choose conditions of cyclic deformation under which the effect of reinforcement appeared to be the most pronounced. The other series involved cyclic tests with a stress-controlled program and relaxation tests. Observations in these experiments were employed to determine adjustable parameters in constitutive equations. Each test was carried out on a new sample and repeated at least two times to assess reproducibility of measurements. Following common practice, stress-controlled tests are defined as cyclic tests in which tensile stress r oscillates between its minimum rmin and maximum rmax values. To compare experimental data in these tests with those in conventional tensile tests and in strain-controlled cyclic tests on the same hybrids [17,40], loading and unloading are performed with constant strain rates. The latter implies that, strictly speaking, the experimental program should be treated as a combination of stress- and strain-controlled experiments. The first series involved four ratcheting tests with various maximum stresses rmax, minimum stresses rmin, and strain rates _ . In each test, a specimen was stretched with a fixed cross-head speed d_ up to a maximum stress rmax, retracted down to a minimum stress rmin with the same cross-head speed, reloaded up to the _ unloaded down maximum stress rmax with the cross-head speed d, min to the minimum stress r with the same cross-head speed, etc. Strain rate _ in a test with cross-head speed d_ was determined as a slope of the straight line that approximated the dependence (t) measured by the extensometer. Observations in cyclic tests on polypropylene/clay nanocomposites are reported in Figs. 1–5. In Fig. 1, experimental stress–strain diagrams are depicted for cyclic tests (50 cycles) with cross-head speed d_ ¼ 100 mm/min (which corresponds to strain rate _ ¼ 1:7  102 s1), maximum stress rmax = 30 MPa, and minimum stress rmin = 10 MPa. This figure demonstrates that reinforcement of polypropylene with nanoclay results in a noticeable decrease in maximum max and minimum min strains per cycle, which is equivalent to a strong growth of fatigue resistance. It is difficult, however, to monitor this phenomenon in detail due to overlapping of the experimental data. To make presentation mode clear, the experimental dependencies max(n) and min(n) are plotted in Fig. 2. This figure shows that the presence of nanoclay results in a reduction of maximum strain at the end of experiment by a factor of 2.5, and this decrease is practically independent of clay content v. Experimental dependencies of maximum and minimum strains per cycle on number of cycles at other experimental conditions are reported in Figs. 3–5. In these figures, observations are presented in cyclic tests (100 cycles) with cross-head speed d_ ¼ 10 mm/min

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Fig. 1. Stress r versus strain . Circles: experimental data in cyclic tests (50 cycles) with cross-head speed 100 mm/min, rmax = 30 MPa, and rmin = 10 MPa on nanocomposites with various clay contents v wt.%. Solid lines: guides for eye.

Fig. 2. Maximum max and minimum min strains per cycle versus number of cycles n. Symbols: experimental data in cyclic tests with cross-head speed 100 mm/min, rmax = 30 MPa, and rmin = 10 MPa on nanocomposites with various clay contents v wt.%.

(which corresponds to strain rate _ ¼ 1:7  103 s1), and parameters rmax = 20 MPa, rmin = 10 MPa (Fig. 3), rmax = 25 MPa, rmin = 15 MPa (Fig. 4), and rmax = 25 MPa, rmin = 20 MPa (Fig. 5). The following conclusions are drawn from Figs. 1–5: 1. The strongest improvement of fatigue resistance is reached when nanoclay content equals 1 wt.%. This result is in accord with our previous studies on the same hybrids, which demonstrated that nanocomposite with v = 1 wt.% had the highest resistance to creep flow in short-term creep tests [17]. 2. The effect of reinforcement on growth of maximum strain per cycle becomes stronger with an increase in rmax. 3. Given a maximum stress rmax, the presence of nanoclay causes more significant resistance to fatigue at higher values of rmin. 4. Growth of cross-head speed leads to more pronounced differences between the mechanical response of neat polypropylene and nanocomposites.

Fig. 3. Maximum max and minimum min strains per cycle versus number of cycles n. Symbols: experimental data in cyclic tests with cross-head speed 10 mm/min, rmax = 20 MPa, and rmin = 10 MPa on nanocomposites with various clay contents v wt.%.

Fig. 4. Maximum max and minimum min strains per cycle versus number of cycles n. Symbols: experimental data in cyclic tests with cross-head speed 10 mm/min, rmax = 25 MPa, and rmin = 15 MPa on nanocomposites with various clay contents v wt.%.

Based on these observations, we focus in what follows on the analysis of nanohybrids with v = 0 and 1 wt.% only. The experiments used for identification of parameters in constitutive equations involved (i) cyclic tests with rmax = 30 MPa, rmin = 20 MPa, d_ ¼ 100 mm/min, and (ii) relaxation tests with strain  = 0.1. Our choice of parameters for cyclic tests was based on the following consideration: (i) rmax = 30 MPa was selected from the condition that fracture of neat polypropylene occured after at least 40 cycles of loading–retraction, (ii) rmin = 20 MPa was chosen as the highest value of rmin at which good approximation of the stress–strain diagram could be performed (when the difference rmax  rmin becomes less than 10 MPa, each path of loading– unloading becomes too short to find material constants with an acceptable accuracy), (iii) d_ ¼ 100 mm/min was chosen due to the fact that at higher cross-head speeds experimental errors in determination of maximum and minimum stresses per cycle became substantial.

A.D. Drozdov, J.deC. Christiansen / Computational Materials Science 53 (2012) 396–408

Fig. 5. Maximum max and minimum min strains per cycle versus number of cycles n. Symbols: experimental data in cyclic tests with cross-head speed 10 mm/min, rmax = 25 MPa, and rmin = 20 MPa on nanocomposites with various clay contents v wt.%.

399

Fig. 7. Maximum max and minimum min strains per cycle versus number of cycles n. Circles: observations in cyclic tests with cross-head speed 100 mm/min, rmax = 30 MPa, and rmin = 20 MPa on nanocomposites with v = 0 and 1 wt.% of clay. Solid lines: results of numerical simulation.

Experimental stress–strain curves for initial 16 cycles of loading–retraction are reported in Fig. 6 (the diagrams for the first two cycles and the last cycle are re-plotted in Fig. 12 together with results of numerical simulation). The experimental dependencies of maximum and minimum strains per cycle on number of cycles n are depicted in Fig. 7 (breakage of samples with v = 0 occurred at the 50th cycle, whereas no signs of failure were observed on specimens with v = 1 wt.% in experiments with 270 cycles). In relaxation tests, specimens were stretched with cross-head speed 100 mm/min up to the required strain . Afterwards, the strain was preserved constant, and a decrease in stress r was monitored as a function of time t. Following the ASTM protocol E–328 for short-term relaxation tests, duration of relaxation tests trel = 20 min was chosen. Observations in relaxation tests are presented in Fig. 8, where stress r is plotted versus relaxation time t0 = t  t0 (t0 stands for

Fig. 8. Stress r versus relaxation time t0 . Symbols: experimental data in relaxation tests with  = 0.1 on nanocomposites with various clay contents v wt.%. Solid lines: results of numerical simulation.

the instant when the relaxation process starts). Following common practice, a semi-logarithmic plot is chosen with log = log10. Fig. 8 shows that the relaxation curves are weakly affected by nanoclay content: the diagrams r(log t0 ) for v = 0 and 1 wt.% are similar to each other. 3. Constitutive model

Fig. 6. Stress r versus strain . Circles: experimental data in cyclic tests (16 cycles) with cross-head speed 100 mm/min, rmax = 30 MPa, and rmin = 20 MPa on nanocomposites with v = 0 and 1 wt.% of clay. Solid lines: results of numerical simulation.

With reference to the homogenization concept, 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 composite. An incompressible, inhomogeneous, transient, non-affine network of polymer chains bridged by junctions is chosen as the equivalent continuum.

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The incompressibility hypothesis is confirmed by our observations in uniaxial tensile tests with cross-head speed 200 mm/ min, in which longitudinal and transverse strains were measured simultaneously. Poisson’s ratios of nanocomposites with v = 0 and 1 wt.% read 0.498 and 0.476, respectively. 3.1. Kinematics of plastic deformations To describe plastic flow in the equivalent medium at small ^ is presented as strains, the strain tensor for macro-deformation  ^e and plastic  ^p strain tensors the sum of elastic 

^ ¼ ^e þ ^p :

ð1Þ

^p is split into the sum of two components The plastic strain tensor  that reflect inelastic deformations in the matrix and inclusions

^pð1Þ þ ^ð2Þ ^p ¼  p :

ð2Þ

The strain rate for plastic deformation in the matrix is proportional to strain rate for macro-deformation ð1Þ d ^p



dt

¼/

^ d : dt

ð3Þ

The non-negative function / in Eq. (3) (i) vanishes in the undeformed state (which means that no plastic deformation occurs in the matrix at very small strains), (ii) monotonically grows under active loading and decreases under retraction (which reflects stressinduced acceleration of plastic flow), and (iii) reaches its ultimate value /1 = 1 at large strains (when the rates of plastic deformation and macro-deformation coincide). Changes in / with time are governed by the differential equation

d/ ¼ að1  /Þ2 _ ; dt

/ð0Þ ¼ 0;

ð4Þ

where the signs ‘‘+’’ correspond to loading and retraction, 1  and ‘‘’’ respectively, _ ¼ 23 ddt^ : ddt^ 2 stands for the equivalent strain rate for macro-deformation, and the coefficient a adopts different values a1 and a2 under active deformation and unloading. For uniaxial tensile cyclic deformation, which is the subject of experimental investigation, loading and unloading processes are determined unambiguously: tensile strain  increases under loading and decreases at unloading. A discussion of loading–unloading processes for an arbitrary three-dimensional cyclic deformation is provided in Section 5.

f ðv Þ ¼ 0

ðv < 0Þ: ð7Þ

An advantage of Eq. (7) is that it involves the only adjustable parameter R > 0. The pre-factor f0 is determined from the normaliR1 zation condition 0 f ðv Þdv ¼ 1. Parameters Na, c, and R are independent of mechanical factors. 3.3. Rearrangement of chains Rearrangement of a temporary network is described by a function n(t, s, v) which equals the number (per unit volume) of temporary chains at time t P 0 that have returned into the active state before instant s 6 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

nðt; t; v Þ ¼ na ðv Þ;

ð8Þ

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). The number of chains that were active at the initial instant and detach from their junctions within the interval [t, t + dt] reads @n/@t(t, 0, v) dt, the number of dangling chains that return into the active state within the interval [s, s + ds] is given by P(s, v) ds with

Pðs; v Þ ¼

  @n ðt; s; v Þ ; @s t¼s

ð9Þ

and the number of chains that merged (for the last time) with the network within the interval [s, s + ds] and detach from their junctions within the interval [t, t + dt] equals @ 2n/@t@ s(t, s, v) dt ds. Detachment of chains from their junctions is described by the kinetic equations

@n ðt; 0; v Þ ¼ Cðv Þnðt; 0; v Þ; @t @n ¼ Cðv Þ ðt; s; v Þ; @s

@2n ðt; s; v Þ @t@ s ð10Þ

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 Eq. (10) with initial conditions (6), (8), and (9) implies that

@n ðt; s; v Þ @s ¼ Na f ðv ÞCðv Þ exp½Cðv Þðt  sÞ:

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 C = cexp[u/(kBT)], where c is an attempt rate, kB denotes Boltzmann’s constant, and T stands for the absolute temperature. Introducing dimensionless activation energy v = u/ (kBT), we present this equation in the form

ð5Þ

Denote by Na concentration of active chains in the equivalent network [42]. The number of active chains na(v) in meso-domains with activation energy v (per unit volume) reads

na ðv Þ ¼ Na f ðv Þ;

ðv P 0Þ;

nðt; 0; v Þ ¼ Na f ðv Þ exp½Cðv Þt;

3.2. Heterogeneity of the network

Cðv Þ ¼ c expðv Þ:

  1  v 2 f ðv Þ ¼ f0 exp  2 R

ð6Þ

where f(v) stands for a distribution function of meso-domains. With reference to the random energy model [43], the quasi-Gaussian expression is accepted for this function

ð11Þ

3.4. Stress–strain relations The strain energy of an active chain reads

w¼

1 l ^e : ^e ; 2

ð12Þ

 stands for rigidity of a chain, and the colon denotes convowhere l lution of tensors. The strain energy of a chain that has last returned into the active state at instant s < t is determined by Eq. (12), where ^ e ðtÞ is replaced with ^e ðtÞ  ^e ðsÞ. The strain energy density (per unit volume) of individual chains in a transient polymer network is given by

1  W 1 ðtÞ ¼ l 2

Z

1

nðt;0; v Þ dv ^e ðtÞ : ^e ðtÞ þ 0  ^e ðsÞÞ ds : ^e ðsÞÞ : ð^e ðtÞ  

Z 0

1

dv

Z

t 0

@n ðt; s; v Þð^e ðtÞ @s ð13Þ

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The first term in Eq. (13) equals the energy of active chains that have not been rearranged within the interval [0, t]. The other term expresses the energy of chains that have last merged with the network at various instants s 2 [0, t] and remained active until instant t. At the nth cycle of cyclic deformation, the energy of interaction between chains and nanofiller is described by the formula similar to Eq. (12)

i 1 h ð2Þ ð2Þ ~ ^p : ^p  ^pð2Þ0 : ^pð2Þ0 ; W2 ¼ l 2

ð14Þ ð2Þ

^p ~ > 0 stands for an analog of elastic modulus, the tensor  where l ^pð2Þ0 describes irreversible deformation in stacks of clay platelets,  ^pð2Þ at the instants when transition from coincides with the tensor  loading to unloading or from unloading to loading occurs, and the ~ accepts different values l ~ 1 and l ~ 2 under loading and coefficient l unloading, respectively. The last term in Eq. (14) does not affect constitutive equations as strain energy is determined up to an arbitrary additive constant. Strain energy density of the equivalent network equals the sum of strain energies of individual chains and the energy of inter-chain interaction

W ¼ W1 þ W2:

ð15Þ

Differentiation of Eq. (15) with respect to time with the use of Eqs. (10), (13), and (14) results in

Z 1 Z 1 Z t dW @n  nðt; 0; v Þ dv ^e ðtÞ þ dv ðt; s; v Þð^e ðtÞ ðtÞ ¼ l dt 0 0 0 @s  ^pð2Þ ^e d d ~ ^ð2Þ ðtÞ þ l ðtÞ  JðtÞ; ð16Þ ^e ðsÞÞ ds : p ðtÞ : dt dt where

JðtÞ ¼

1 l 2

Z

1



Cðv Þ dv nðt; 0; v Þ^e ðtÞ : ^e ðtÞ

0

 ^e ðsÞÞ ds P 0: ^e ðsÞÞ : ð^e ðtÞ  

Z 0

t

@n ðt; s; v Þð^e ðtÞ @s

  Z 1 Z t dW @n  Na ^e ðtÞ  dv ðt; s; v Þ^e ðsÞ ds ðtÞ ¼ l dt 0 0 @s " # ð2Þ ^ ^ dp d ~ ^ð2Þ ðtÞ þ l : ð1  /ðtÞÞ ðtÞ  p ðtÞ dt dt ð2Þ

^p d ðtÞ  JðtÞ: dt

ð17Þ

For isothermal volume-preserving deformation, the Clausius– Duhem inequality reads

dQ dW d^ ^0 : ¼ þr P 0; dt dt dt

ð18Þ

^ 0 dewhere Q stands for internal dissipation per unit volume, and r ^ . Combining Eqs. (17) and notes deviatoric part of the stress tensor r (18) and using Eq. (11), we arrive at the stress–strain relation

r^ ðtÞ ¼ pðtÞbI þ lð1

 Z  /ðtÞÞ ^e ðtÞ  0

1

f ðv Þ dv

Z

  Z 1 dQ b v Þdv ðtÞ ¼ JðtÞ þ l ^e ðtÞ  R^ð2Þ ðtÞ  f ð v Þ Zðt; p dt 0 :

^ð2Þ d p ðtÞ P 0; dt

ð20Þ

where R adopts the values

R1 ¼

l~ 1 ; l

R2 ¼

l~ 2 l

ð21Þ

b vÞ ¼ under loading and retraction, and the function Zðt; Rt ^ C ð v Þ expð C ð v Þðt  s ÞÞ  ð s Þ d s obeys the differential equation e 0

b @Z b v Þ; ðt; v Þ ¼ Cðv Þ½^e ðtÞ  Zðt; @t

b v Þ ¼ 0: Zð0;

ð22Þ

The Clausius–Duhem inequality (20) is satisfied provided that

  Z 1 ^ð2Þ d p b v Þdv ; f ðv Þ Zðt; ðtÞ ¼ S_ ^e ðtÞ  R^ð2Þ p ðtÞ  dt 0

ð23Þ

where the coefficient S adopts different (but constant) values, S1 and S2, under loading and unloading. Stress–strain relation (19) together with kinematic Eqs. (1) and (2) and kinetic Eqs. (3), (4), (22), and (23) provide constitutive equations for the viscoelastoplastic behavior of polymer nanocomposites under cyclic deformation. These equations involve three material constants, l, c, R, and six adjustable functions, a1, a2, R1, R2, S1, S2, with the following physical meaning:  l stands for an elastic modulus of a polymer nanocomposite,  c and R characterize its linear viscoelastic behavior,  a1 and a2 describe irreversible deformation in the matrix under loading and unloading,  R1, S1 and R2, S2 characterize plastic flow in stacks of clay platelets under active deformation and retraction. 3.5. Adjustable functions

Transformation of Eq. (16) by means of Eqs. (1), (2), (3), (6), and (8) yields

:

It follows from Eqs. (18) and (19) that

 t Cðv ÞexpðCðv Þðt  sÞÞ^e ðsÞ ds ;

0

ð19Þ  N a ; p is an unknown pressure, and bI stands for the unit where l ¼ l tensor.

Evolution of material functions under cyclic deformation is described by the following scenario: 1. The coefficients S1 and S2 are rapidly equilibrated within the first two cycles. For active loading of a virgin specimen, S1 = 0, while for subsequent reloadings, S1 = S11. Similarly, S2 adopts some value S20 for the first retraction, whereas for subsequent unloadings, S2 = S21. Three constants, S20, S11, and S21, entirely determine these coefficients. 2. The coefficients a1 and a2 are equilibrated within a few (5–6) initial cycles. In subsequent cycles of loading–retraction, they adopt their ultimate values a11 = a21 = a1. 3. The coefficient R2 is function of intensity of  a decreasing 1 ^p 2 at instants when transition ^p :  plastic strain p ¼ 23   . This occurs from active loading to unloading p ¼ max p function obeys the differential equation

dR2 ¼ A2 R22 ; dmax p

ð24Þ

where A2 is a positive coefficient. 4. The coefficient R1 is split into the sum of two components

R1 ¼ r þ R;

ð25Þ

where evolution of r is induced by damage accumulation in stacks of clay platelets, and R characterizes changes in energy of inter-particle interaction driven by plastic flow.

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5. For a cyclic deformation program with fixed maximum and minimum stresses, number of cycles n serves as a measure of damage accumulation. The effect of this quantity on r is described by the equation

r ¼ r 0 expðanÞ;

ð26Þ

where r0 and a are positive constants. The decay in r with n occurs rather rapidly, which implies that r vanishes after a transition period (about 10 cycles). For periodic loading programs with time-dependent amplitudes and frequencies of oscillations, n should be replaced in Eq. (26) with some internal variable that reflects growth of material damage under cyclic deformation [44]. 6. A decrease in R with plastic strain is governed by the differential equation

dR ¼ A1 ðR1  RÞ; dmin p

ð27Þ

where A1 and R1 are adjustable parameters, and min stands p for intensity of plastic strain p at instants when transition occurs from retraction to reloading. After an initial transition period, the viscoelastoplastic response of a polymer/nanoclay hybrid is determined by 10 material constants: l, c, R, A1, A2, a1, r1, R1, S11, and S21. Although this number is not small, it is substantially lower than the number of adjustable parameters in conventional constitutive models for multi-cycle loading [37–39]. An important advantage of the present approach is that the material constants are found by fitting loading and unloading paths of stress–strain diagrams step by step, which implies that not more than three parameters are determined by matching each set of observations.

4. Fitting of observations To find adjustable parameters in the constitutive equations, we approximate the experimental data depicted in Figs. 6–8. Each set of observations is matched separately. 4.1. Uniaxial tension We begin with simplification of the stress–strain relations for uniaxial deformation of an incompressible medium with

  1 ^ ¼  e1 e1  ðe2 e2 þ e3 e3 Þ ; 2

ð28Þ

where  = (t) stands for tensile strain, and e1, e2, e3 denote unit vectors of a Cartesian coordinate frame. The strain tensors ^ e ; ^p ; ^pð1Þ , and ^pð2Þ are determined by Eq. (28) with coefficients e ; p ; pð1Þ , and pð2Þ , respectively. It follows from Eqs. (1), (2), (3) and (28) that

e ¼   p ; where

pð1Þ

p ¼ pð1Þ þ ð2Þ p ;

satisfies the differential equation

ð1Þ

dp ¼ /_ dt

ð29Þ

ð1Þ

ðloadingÞ;

dp ¼ /_ dt

ðunloadingÞ:

ðloadingÞ; ðunloadingÞ:

obeys Eq. (23)

 Z 1 d f ðv ÞZðt; v Þdv ðloadingÞ; ¼ S1 _ e ðtÞ  R1 ð2Þ p ðtÞ  dt 0   Z 1 ð2Þ dp ðunloadingÞ: ð32Þ ðtÞ  f ð v ÞZðt; v Þd v ¼ S2 _ e ðtÞ  R2 ð2Þ p dt 0 The function Z(t, v) satisfies Eq. (22)

@Z ðt; v Þ ¼ Cðv Þ½e ðtÞ  Zðt; v Þ; @t

ð33Þ

where C(v) is given by Eq. (5). The engineering stress r reads



rðtÞ ¼ Eð1  /ðtÞÞ e ðtÞ 

Z

1

 f ðv ÞZðt; v Þdv ;

ð34Þ

0

where E ¼ 32 l stands for the Young’s modulus. Eqs. (29)–(34) provide stress–strain relations for uniaxial tension of a specimen with an arbitrary deformation program (t). 4.2. Relaxation tests For tensile relaxation test with a strain , Eqs. (29)–(33) imply that e, p, and / remain constant. It follows from Eqs. (5) and (34) that

rðtÞ ¼ r0

Z

1

f ðv Þ expðct expðv ÞÞ dv ;

ð35Þ

0

where r0 = E(1  /)e. Adjustable parameters c and R are determined by fitting the observations reported in Fig. 8. We fix some intervals [0, c°] and [0, R°], where c and R are located, and divide these intervals into J = 10 sub-intervals by the points c(i) = iDc, R(j) = jDR with Dc = c°/J, DR = R°/J (i, j = 0, 1, . . . , J  1). For each pair {c(i), R(j)}, stress r is found from Eq. (35), where the integral is calculated by the Simpson method with v = mDv, Dv = 2.0, m = 0, 1, . . . , M, and M = 20. The coefficient r0 is determined by the least-squares technique from the condition of minimum of the function

F1 ¼

X ½rexp ðt k Þ  rnum ðt k Þ2 ; k

where summation is performed over all instants tk at which the observations are presented, rexp stands for the stress measured in the test, and rnum is given by Eq. (35). The quantities c and R are found from the condition of minimum of function F1. Afterwards, the initial intervals are replaced with the new intervals [c  Dc, c + Dc] and [R  DR, R + DR], and the calculations are repeated. Fig. 8 demonstrates good agreement between the observations in relaxation tests and the results of numerical simulation. Material constants c and R are listed in Table 1. These parameters are close to c and R reported in [17], where observations were presented in relaxation tests with smaller strains. The latter may serve as a justification of our neglect of nonlinear viscoelastic phenomena (the model disregards the effect of strain on rate of relaxation c). Table 1 Adjustable parameters for polypropylene/nanoclay hybrids. Parameter

v=0

v=1

E (GPa) c (s1)

2.04 0.10 10.9 4.70 12.9 19.6 9.13 51.5 6.84 58.0

2.48 0.34 11.7 14.4 12.7 26.5 9.56 73.0 2.53 44.1

R

ð31Þ

S11 S20 S21 a1 A1 R1 A2

d/ dt

ð2Þ p 

ð2Þ p

ð30Þ

The function / is governed by Eq. (4)

d/ ¼ a1 _ ð1  /Þ2 dt ¼ a2 _ ð1  /Þ2

The strain

A.D. Drozdov, J.deC. Christiansen / Computational Materials Science 53 (2012) 396–408

403

4.3. First loading

(33), (34), and (37) are integrated numerically from  = max to  = min. The coefficients R2 and S2 are calculated from the condition

Under tension of a virgin sample, stress–strain relations (30)– (32) are given by

of minimum of function F2. Afterwards, the initial intervals are replaced with new intervals [R2  DR, R2 + DR] and [S2  DS, S2 + DS], and the calculations are repeated. The best-fit value S20 of S2 is listed in Table 1, and the best-fit parameter R2 is depicted in Fig. 10.

ð1Þ

dp ¼ /_ ; dt

ð2Þ

dp ¼ 0; dt

d/ ¼ a1 _ ð1  /Þ2 : dt

ð36Þ

Adjustable parameters E and a1 in Eqs. (29), (33), (34), and (36) are determined by fitting the observations reported in Fig. 6. We fix an interval [0, a°], where a1 is located, and divide this interval into J = 10 sub-intervals by the points a(i) = iDa with Da = a°/J (i = 0, 1, . . . , J  1). For each {a(i)}, Eqs. (29), (33), (34), and (36) are integrated numerically from  = 0 to  = max, where max is the maximum strain under stretching. Integration over time is performed by the Runge–Kutta method with step Dt = 0.001 s. The integral in Eq. (34) is calculated by the Simpson method. The Young’s modulus E is determined by the least-squares technique from the condition of minimum of the function

F2 ¼

X ½rexp ðk Þ  rnum ðk Þ2 ; k

where summation is performed over all strains k at which the observations are presented, rexp is the stress measured in an appropriate test, and rnum is given by Eq. (34). The coefficient a1 is calculated from the condition of minimum of function F2. Afterwards, the initial interval is replaced with the new interval [a1  Da, a1 + Da], and the calculations are repeated. The best-fit elastic modulus E is given in Table 1, and the coefficient a1 is reported in Fig. 9. 4.3.1. First unloading Under first retraction, stress–strain relations (30)–(32) read ð1Þ

ð2Þ

dp dp ¼ /_ ; ¼ S2 _ dt dt d/ ¼ a2 _ ð1  /Þ2 : dt



e ðtÞ  R2 pð2Þ ðtÞ 

Z

1



f ðv ÞZðt; v Þdv ;

4.4. First reloading Under first reloading, stress–strain relations (30)–(32) are presented in the form

  Z 1 ð1Þ ð2Þ dp dp f ðv ÞZðt; v Þdv ; ¼ /_ ; ¼ S1 _ e ðtÞ  R1 pð2Þ ðtÞ  dt dt 0 d/ 2 ¼ a1 _ ð1  /Þ : ð38Þ dt To determine adjustable parameters a1, R1, and S1 in Eqs. (29), (33), (34), and (38), we fix some intervals [0, a°], [0, R°], and [0, S°], where a1, R1, and S1 are located, and divide these intervals into J = 10 subintervals by the points a(i) = iDa, R(j) = jDR, S(l) = lDS with Da = a°/J, D R = R°/J, DS = S°/J (i, j, l = 0, 1, . . . , J  1). For each triplet {a(i), R(j), S(l)}, Eqs. (29), (33), (34), and (38) are integrated numerically from  = min to  = max. The coefficients a1, R1, and S1 are found from the condition of minimum of function F2. Afterwards, the initial intervals are replaced with new intervals [a1  Da, a1 + Da], [R1  DR, R1 + DR], [S1  DS, S1 + DS], and the calculations are repeated. The best-fit value S11 of S1 is reported in Table 1. The best-fit coefficients a1 and R1 are presented in Figs. 9 and 10, respectively. 4.5. Subsequent unloadings and reloadings

To reduce the number of parameters to be determined, we set a2 = 0 and find R2 and S2 in Eqs. (29), (33), (34), and (37) by fitting the observations depicted in Fig. 6. We fix some intervals [0, R°] and [0, S°], where R2 and S2 are located, and divide these intervals into J = 10 sub-intervals by the points R(i) = iDR, S(j) = jDS with DR = R°/J, DS = S°/J (i, j = 0, 1, . . . , J  1). For each pair {R(i), S(j)}, Eqs. (29),

Each path (reloading and unloading) of the stress–strain curves depicted in Fig. 6 is matched separately by means of the above algorithms with two adjustable parameters: a1 and R1 for stretching, and a2 and R2 for retraction. The second unloading provides the only exception from this rule: in this case, three adjustable parameters, a2, R2, S2, are employed. For subsequent unloading paths, the value S2 = S21 found by matching the second retraction path is used (this value is reported in Table 1). Fig. 6 demonstrates good agreement between the observations in cyclic tests and the results of numerical analysis. To reveal the

Fig. 9. Parameters a1 () and a2 () versus number of cycles n. Circles: treatment of observations in cyclic tests with rmax = 30 MPa and rmin = 20 MPa on nanocomposites with v = 0 and v = 1 wt.% of clay. Solid lines: approximation of the data by constants.

Fig. 10. Parameter R2 versus plastic strain max . Circles: treatment of observations p in cyclic tests with rmax = 30 MPa and rmin = 20 MPa on nanocomposites with v = 0 and v = 1 wt.% of clay. Solid lines: approximation of the data by Eq. (24).

0

ð37Þ

404

A.D. Drozdov, J.deC. Christiansen / Computational Materials Science 53 (2012) 396–408

quality of fitting, the experimental stress–strain curves for the first two cycles and the last (16th) cycle are reported in Fig. 12 together with the results of simulation. This figure shows that accuracy of approximation is not reduced with an increase in number of cycles. The dependencies a1(n) and a2(n) are reported in Fig. 9. This figure demonstrates that after a transition period (about 5–6 cycles), a1 and a2 coincide and become independent of n, in accord with the scenario of Section 3.5. The ultimate values a1 are found by matching observations in cycles from 7 to 16 with constants. These values are listed in Table 1. The dependencies of R2 on maximum plastic strain per cycle max (the latter quantity is found by numerical integration of the p stress–strain relations) are depicted in Fig. 10. The data are approximated by Eq. (24), where the coefficient A2 (listed in Table 1) is found by means of the following algorithm. The general solution of Eq. (24) reads

1 ¼ C 2 þ A2 max p ; R2

ð39Þ

where C2 is a constant. Parameters A2 and C2 are calculated by the max least-squares method when R1 (Fig. 13). 2 is plotted versus p Coefficients A1 and R1 in Eq. (27) are determined by matching the observations depicted in Fig. 7. For this purpose, we fix some intervals [0, A°] and [0, R°], where A1 and R1 are located, and divide these intervals into J = 10 sub-intervals by the points A(i) = iDA, R(j) = jDR with DA = A°/J, DR = R°/J (i, j = 0, 1, . . . , J  1). For each pair {A(i), R(j)}, Eqs. (27), (29), (30), (31), (32), (33), (34) are integrated numerically with the adjustable parameters depicted in Figs. 9– 11 and collected in Table 1. The coefficients A1 and R1 are found from the condition of minimum the function

F3 ¼

Xh ðmax

exp

ðnÞ  max

num

2

ðnÞÞ þ



min

exp

ðnÞ  min

num

Fig. 12. Stress r versus strain . Circles: experimental data in cyclic tests (the first two cycles and the 16th cycle) with rmax = 30 MPa and rmin = 20 MPa on nanocomposites with v = 0 and v = 1 wt.% of clay. Solid lines: results of numerical simulation.

2 i ðnÞ ;

n

where summation is performed over all n at which the data are presented, max, min are maximum and minimum strains measured in an appropriate test, and max num, min num are calculated in numerical simulation of the constitutive equations. The best-fit parameters A1 and R1 are reported in Table 1. Appropriate solutions of Eq. (27) are plotted in Fig. 11 together with the experimental dependencies R1 ðmin p Þ. Fig. 11 reveals fair agreement between the

max Fig. 13. Parameter R1 . Circles: treatment of observations 2 versus plastic strain p in cyclic tests with rmax = 30 MPa and rmin = 20 MPa on nanocomposites with v = 0 and v = 1 wt.% of clay. Solid lines: approximation of the data by Eq. (39).

data found in the fitting procedure and their fits by Eq. (27) after an initial transition period (5–6 cycles). To describe evolution of R1 within the transition period, the parameter r is calculated from Eq. (25), where R is given by Eq. (27). The dependence r(n) is depicted in Fig. 14. The data are approximated by Eq. (26), where a is found by the method of nonlinear regression, and r0 is determined by the least-squares technique. Fig. 14 confirms that r vanishes after 5–6 cycles of periodic loading, in accord with the scenario of Section 3.5. 5. Discussion

Fig. 11. Parameter R1 versus plastic strain min p . Circles: treatment of observations in cyclic tests with rmax = 30 MPa and rmin = 20 MPa on nanocomposites with v = 0 and v = 1 wt.% of clay. Solid lines: approximation of the data by Eq. (27).

A constitutive model is developed for the viscoelastic and viscoplastic behavior of polymer nanocomposites under cyclic deformations, and adjustable parameters in the stress–strain relations are found by fitting observations (16 cycles of loading–unloading) in tests with a stress-controlled program (ratcheting). An important

A.D. Drozdov, J.deC. Christiansen / Computational Materials Science 53 (2012) 396–408

advantage of the model is that it correctly describes experimental dependencies of maximum max and minimum min strains per cycle on number of cycles n. We intend now to concentrate of the following questions: 1. What may be a physical mechanism for substantial reduction in ratcheting strain when polypropylene is reinforced with nanoclay? 2. Is it possible to predict fatigue failure of nanocomposites by using experimental stress–strain diagrams during first several cycles with reasonable accuracy? 3. How the constitutive equations may be applied to the analysis of three-dimensional cyclic deformations, for which definitions of loading and unloading processes become non-obvious? 5.1. Enhancement of resistance to fatigue in nanocomposites To provide an answer to the first question, we compare adjustable parameters collected in Table 1. According to this table, reinforcement of polypropylene with 1 wt.% of nanoclay results in an increase in Young’s modulus E by 22%, which reflects improvement of its elastic properties. The effect of nanofiller on viscoelastic properties appears to be of secondary importance. It is observed as a weak broadening of the relaxation spectrum (R grows by 7%) and an increase in the rate of rearrangement c (by 3 times). The growth of relaxation rate does not seem dramatic, as log c is conventionally employed in the Eyring or Arrhenius relations. The equilibrium rate of viscoplastic flow in the polymer matrix a1 remains practically unaffected by clay content (it increases by less than 5% due to the presence of nanoclay). Rather substantial changes are observed in coefficients S11 and ^ð2Þ S21 that reflect evolution of plastic strain tensor  p . This strain tensor was introduced in Section 3 as a measure of irreversible deformations in inclusions. For a polymer/nanoclay hybrid with a semicrystalline matrix, inclusions may be associated with stacks of clay platelets, as well as with the crystalline phase of the polymer. Reinforcement of polypropylene with 1 wt.% of nanoclay results in an increase in S11 by 3 times and in S21 by 35%. Keeping in mind that S1 and S2 are proportional to rates of plastic flow under loading and unloading, one can conclude that the effect of filler is stronger at loading than at retraction.

405

This result is confirmed by comparison of ultimate values R1 of parameter R1. According to Eqs. (14) and (21), R1 serves as a measure of energy of interaction between polymer chains and inclusions under loading. A pronounced decay in R1 (R1 at v = 0 exceeds that at v = 1 wt.% by a factor of 2.7) may reflect destruction of stacks of clay platelets under cyclic deformation. The coefficients A1 and A2 that describe decay of R1 and R2 with plastic strain p adopt similar values (A1 at v = 1 exceeds that at v = 0 wt.% by 42%, while A2 at v = 0 is higher than A2 at v = 1 wt.% by 32%). However, if the decrease in R1 and R2 is analyzed as functions of n, the situation changes pronouncedly: after 16 cycles of loading–retraction, R1 at v = 0 exceeds that at v = 1 wt.% by a factor of 3.3, whereas R2 at v = 1 exceeds that at v = 0 wt.% by a factor of 2.5. These estimates lead to the conclusion that a strong increase in fatigue resistance due to reinforcement (revealed in Figs. 1–7) may be ascribed to (i) a substantial growth of rates of plastic flow in inclusions which results in (ii) dramatic reduction in the energy of interaction between polymer chains and inclusions. 5.2. Prediction of fatigue failure To adequately describe the experimental curves max(n) and min(n) depicted in Fig. 7, these dependencies were fitted by using two adjustable parameters, A1 and R1. Fig. 11 demonstrates that this procedure was not necessary, as the same coefficients could be found by approximating the data that describe evolution of R1 with min p . This implies that, in principle, growth of maximum and minimum strains per cycle can be predicted by using experimental stress–strain diagrams measured during the first several cycles of loading–unloading. However, to reach an acceptable accuracy of prediction, the number of cycles should not be too small, as changes in R1 driven by damage accumulation within a transition period (they are described by the dependencies r(n) reported in Fig. 14) should be excluded from the consideration. As a rule of thumb, 15–20 cycles of loading–unloading may be suggested as a reasonable number of oscillations needed for qualitative prediction of ratcheting strain. To examine whether the model is applicable when the number of cycles is typical of conventional low-cycle fatigue tests (which means that n is of order of 103–105), simulation is performed of the stress–strain relations for tensile cyclic deformations with rmax = 30 MPa, rmin = 20 MPa, and various strain rates _ ranging from 102 to 1 s1. Numerical integration of the constitutive equations is conducted with time step Dt = 105 s and the material constants listed in Table 1 (these quantities are presumed to be independent of strain rate). Results of the analysis for tests with 1000 cycles are depicted in Figs. 15 and 16, where max and min are plotted versus n. These figures show a noticeable decrease in maximum and minimum strains per cycle with growth of strain rate. For example, for neat polypropylene, max reaches its critical value ⁄ = 0.2 (at this strain, necking occurs under uniaxial tension with constant strains rates) after 43 cycles at _ ¼ 0:01 s1, 324 cycles at _ ¼ 0:05 s1, and 661 cycles _ ¼ 0:1 s1. Adopting the equality

max ¼ 

ð40Þ

as a condition of fatigue fracture, the number of cycles to failure nf is calculated as a function of strain rate _ . Results of simulation are depicted in Fig. 17, where nf is plotted versus _ in double-logarithmic coordinates. This figure shows that the data are correctly approximated by the equations Fig. 14. Parameter r versus number of cycles n. Circles: treatment of observations in cyclic tests with rmax = 30 MPa and rmin = 20 MPa on nanocomposites with v = 0 and v = 1 wt.% of clay. Solid lines: approximation of the data by Eq. (26).

log nf ¼ c0 þ c1 log _ ; where c0 and c1 are determined by the least-squares method.

ð41Þ

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Fig. 15. Maximum max and minimum min strains per cycle cycles n. Symbols: results of numerical simulation for rmax = 30 MPa, rmin = 20 MPa, and various strain rates _ s1 _ ¼ 5:0  102 ; ⁄ _ ¼ 1:0  101 ; q _ ¼ 5:0  101 ; } _ ¼ 1:0) with v = 0 wt.% of clay.

versus number of cyclic tests with ( _ ¼ 1:0  102 ; on nanocomposite

Fig. 17. Number of cycles to failure nf versus strain rate _ . Circles: results of numerical simulation for cyclic tests with rmax = 30 MPa, rmin = 20 MPa, and fracture criterion max = 0.2 on nanocomposites with v = 0 and v = 1 wt.% of clay. Solid lines: approximation of the data by Eq. (41).

5.3. Loading–unloading conditions Conventional definition of loading and unloading paths of a stress–strain diagram under an arbitrary three-dimensional cyclic deformation is grounded on the concept of yield surface that evolves under loading [45–47]. For an incompressible medium with kinematic and isotropic hardening, this surface is described ^ 0 Þ  g, where a ^ 0 denotes a ^ 0; a by the equation F = 0 with F ¼ f ðr traceless tensor of internal (back) stresses (kinematic hardening), and g > 0 is a scalar function (isotropic hardening). Loading and unloading paths of a stress–strain diagram are determined by the conditions

^ 0 @F dr > 0 ðloadingÞ; : ^0 dt @ r

^ 0 @F dr < 0 ðunloadingÞ: : ^0 dt @ r

For the von Mises function f, these inequalities read

^0 dr ^ 0 Þ > 0 ðloadingÞ; ^0  a : ðr dt < 0 ðunloadingÞ: Fig. 16. Maximum max and minimum min strains per cycle cycles n. Symbols: results of numerical simulation for rmax = 30 MPa, rmin = 20 MPa, and various strain rates _ s1 _ ¼ 5:0  102 ; ⁄ _ ¼ 1:0  101 ; q _ ¼ 5:0  101 ; } _ ¼ 1:0) with v = 1 wt.% of clay.

versus number of cyclic tests with ( _ ¼ 1:0  102 ; on nanocomposite

max ¼ p p

ð42Þ

for all strain rates under consideration (this parameter varied in the interval between 0.132 and 0.135), which implies that fracture criteria (40) and (42) are equivalent, (iii) number of cycles to failure for nanocomposite with v = 1 wt.% of clay exceeds that for neat polypropylene by an order of magnitude.

ð43Þ

The simplest way to establish correlations between our constitutive equations (that do not involve back stresses) and criterion (43) is to ^ 0 ¼ 0 in Eq. (43), we arrive disregard kinematic hardening. Setting a at the condition

^0 dr ^ 0 > 0 ðloadingÞ; :r dt

Three conclusions are worth to be mentioned: (i) the coefficient c1 is rather close to unity (c1 = 0.99 at v = 0 and c1 = 0.71 at v = 1 wt.%), which means that number of cycles to failure is linearly proportional to strain rate, (ii) failure occurs practically at the same value of maximum plastic strain

^0 dr ^0Þ ^0  a : ðr dt

^0 dr ^ 0 < 0 ðunloadingÞ: :r dt

ð44Þ

To demonstrate that the constitutive equations derived in Section 3 are compatible with Eq. (44), numerical integration is performed of the stress–strain relations for two deformation programs: (i) cyclic loading with strain rate _ ¼ 1:7  103 s1 with a stress-controlled program (oscillations between maximum stress rmax = 25 MPa and minimum stress rmin = 0 MPa), and (ii) cyclic deformation with the same strain rate with a symmetric straincontrolled program (oscillations between maximum strain max = 0.036 that corresponds to rmax = 25 MPa and minimum strain min = max). For the stress-controlled program, Eq. (44) coincides with the loading–unloading criterion used in approximation of experimental data. For the strain-controlled program, 1 cycle of oscillations involve: tension up to the maximum strain max (loading-I),

A.D. Drozdov, J.deC. Christiansen / Computational Materials Science 53 (2012) 396–408

.

Table 2 Adjustable parameters for polypropylene.

a1 R1 S1

Loading-I

Loading-II

Loading-III

23.5 0.0 0.0

15.5 7.0 3.3

9.3 0.0 0.0

407

a2 R2 S2

Unloading-I

Unloading-II

17.0 3.1 17.0

12.8 2.3 15.7

The stress–strain curve in Fig. 18 is in qualitatively agreement with the curves depicted in Figs. 1 and 6. The stress–strain diagram plotted in Fig. 19 is similar to observations on other semicrystalline polymers reported in [48–50]. The following conclusions are drawn from Figs. 18 and 19: (i) results of numerical simulation for stress- and strain-controlled cyclic tests based on the constitutive equations derived in Section 3 together with loading–unloading condition (44) are in qualitative agreement with observations, and (ii) the model can describe the Bauschinger effect in semicrystalline polymers observed as a difference between maximum stresses reached under loading and unloading [50].

6. Concluding remarks

Fig. 18. Stress r versus strain simulation for a cyclic test rmin = 0 MPa.

.

Circles and solid line: results of numerical with _ ¼ 1:7  103 s1, rmax = 25 MPa, and

Fig. 19. Stress r versus strain . Circles and solid line: results of numerical simulation for a cyclic test with _ ¼ 1:7  103 s1 and max = min = 0.036.

retraction down to the zero tensile stress (unloading-I), retraction down to the minimum strain max (loading-II), tension up to the zero stress (unloading-II), and stretching up to max (loading-III). As experimental data on polypropylene in cyclic tests with negative min are not available, simulation is conducted with material constants collected in Table 1 and hypothetical adjustable parameters listed in Table 2 (these quantities are similar to the coefficients depicted in Figs. 9–11). Results of numerical analysis are presented in Figs. 18 and 19, where stress r is plotted versus strain

Experimental data are reported on polypropylene/nanoclay hybrids with various concentrations of filler in tensile cyclic tests at room temperature (ratcheting between fixed minimum stresses rmin and maximum stresses rmax). Observations show that reinforcement of polypropylene with nanoclay results in strong (by several times) decrease in ratcheting strain. Substantial enhancement of fatigue resistance is observed at all clay contents and for all experimental conditions. The most pronounced improvement of mechanical properties is reached when concentration of nanoclay equals 1 wt.% at the highest value of maximum stress rmax = 30 MPa. To rationalize these findings, a constitutive model is developed in cyclic viscoelasticity and viscoplasticity of polymer nanocomposites. With reference to the homogenization concept, a nanohybrid is treated as an equivalent polymer network with two features: (i) the plastic strain tensor is split into the sum of two components that reflect irreversible deformations in the polymer matrix and inclusions and obey different flow rules, and (ii) the strain energy density equals the sum of strain energies of individual chains and the energy of interaction between chains and stacks of clay platelets. Stress–strain relations are derived by using the Clausius– Duhem inequality for an arbitrary three-dimensional deformation with small strains. Adjustable parameters in the constitutive equations are found by fitting the experimental data (for each set of observations, 16 cycles of loading–unloading). It is demonstrated that the model correctly describes the stress–strain diagrams and the dependencies of maximum max and minimum min strains per cycle on number of cycles n. Comparison of material constants for neat polypropylene and polymer/nanoclay hybrid leads to a conclusion that enhancement of fatigue resistance may be ascribed to acceleration of plastic flow in clusters of clay platelets which induces noticeable reduction in the energy of interaction between polymer chains and inclusions. Numerical simulation demonstrates that the constitutive equations can be applied to predict fatigue failure of polymer nanocomposites. It is shown that number of cycles to failure nf for the nanocomposite with 1 wt.% of clay exceeds that for neat polypropylene by an order of magnitude in a rather wide (2 decades) interval of strain rates. It is revealed that the constitutive model together with loading– unloading condition (44) can capture qualitatively some features of the mechanical response of semicrystalline polymers in tension– compression cyclic tests, in particular, the Bauschinger effect.

Acknowledgment Financial support by the European Commission through project Nanotough–213436 is gratefully acknowledged.

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