Multi-cycle deformation of semicrystalline polymers: Observations and constitutive modeling

Mechanics Research Communications 48 (2013) 70–75

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Multi-cycle deformation of semicrystalline polymers: Observations and constitutive modeling A.D. Drozdov a,b,∗ , R. Klitkou b , 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 10 February 2012 Received in revised form 30 October 2012 Available online 11 January 2013 Keywords: Polypropylene Viscoelasticity Viscoplasticity Cyclic deformation Fading memory

a b s t r a c t Experimental data are reported on isotactic polypropylene in multi-cycle uniaxial tensile tests where a specimen is stretched up to some maximum strain and retracted down to the zero minimum stress, while maximum strains increase with number of cycles. Fading memory of deformation history is observed: when two samples are subjected to loading programs that differ along the first n − 1 cycles only, their stress–strain diagrams coincide starting from the nth cycle. Constitutive equations are developed in cyclic viscoelasticity and viscoplasticity of semicrystalline polymers, and adjustable parameters in the stress–strain relations are found by fitting the experimental data. Results of numerical simulation demonstrate that the model predicts the fading memory effect quantitatively. To confirm that the observed phenomenon is typical of semicrystalline polymers, experimental data are presented in tensile cyclic tests with large strains on low density polyethylene and compressive cyclic tests on poly(oxymethylene). © 2013 Elsevier Ltd. All rights reserved.

1. Introduction This paper focuses on the experimental investigation and constitutive modeling of the mechanical response of polypropylene in multi-cycle uniaxial tensile tests at room temperature. A mixed deformation program is chosen for experimental study: at each cycle of oscillations, a specimen is stretched up to a maximum strain and retracted down to the zero minimum stress, while maximum strains increase monotonically with number of cycles: εmax < εmax < · · · < εmax . An interesting phenomenon is revealed: N 1 2 when two specimens are subjected to deformation programs which differ along the first n − 1 cycles and coincide afterwards, the stress–strain diagrams of these samples coincide entirely starting from the nth cycle. In other words, when a sample reaches some maximum strain εmax under cyclic deformation, its memory n of deformation history is lost entirely. As a counterpart for the fading memory phenomenon, we would like to mention the Mullins effect (Mullins, 1947; Diani et al., 2009) that is characterized by the following features: (i) loading and unloading paths of a stress–strain diagram differ pronouncedly

∗ Corresponding author at: Department of Plastics Technology, Danish Technological Institute, Gregersensvej 1, Taastrup 2630, Denmark. Tel.: +45 72 20 31 42; fax: +45 72 20 31 12. E-mail address: [email protected] (A.D. Drozdov). 0093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2013.01.001

(hysteresis of energy), (ii) when a sample is subjected to cyclic deformation with fixed maximum strain and minimum stress, maximum stress per cycle decreases monotonically with number of cycles (cyclic softening), and (iii) when a specimen suffered several cycles of loading–unloading with a fixed maximum strain is stretched above this strain, its stress–strain curve reaches rapidly that of a virgin (not subjected to cyclic loading) sample. The difference between our findings and observations mentioned in (iii) is that not only loading paths of the corresponding stress–strain diagrams become identical, but their retraction and reloading paths coincide as well. A conventional approach to modeling the Mullins phenomenon is grounded on the assumption that a polymer possesses a set of different strain energy densities and chooses an appropriate strain energy from this set at the instant when reloading starts (Ogden and Roxburgh, 1999). The choice is presumed to be determined by the amount of damage accumulated under previous deformation (pseudo-elasticity). Applicability of this concept to the analysis of the response of semicrystalline polymers seems questionable: unlike elastomers that demonstrate a small permanent set (which implies that irreversible deformations may be disregarded, at least, as a first approximation), plastic strains in semicrystalline polymers (defined as strains measured when tensile stress vanishes at retraction) are comparable with maximum strains under stretching. The latter makes derivation of a constitutive model challenging: on one hand, the stress–strain relations should account for fading

A.D. Drozdov et al. / Mechanics Research Communications 48 (2013) 70–75

memory of deformation history observed in experiments, while on the other, some memory should be preserved by the model in terms of irreversible strains. The objective of this work is threefold: 1. To report experimental data on polypropylene in uniaxial tensile cyclic tests with increasing maximum strains that reveal the fading memory phenomenon. 2. To develop a constitutive model in cyclic viscoelastoplasticity of semicrystalline polymers and to find its adjustable parameters by fitting the observations. 3. To confirm ability of the model to predict fading memory of deformation history by numerical simulation. Although constitutive modeling of the viscoelastic and viscoplastic responses of polymers subjected to cyclic loading has been a focus of attention in the past decade, most of previous studies (Avanzini, 2008; Drozdov and Christiansen, 2008; Dusunceli and Colak, 2008; Mizuno and Sanomura, 2009; Ayoub et al., 2010; Silberstein and Boyce, 2010; Hassan et al., 2011; Hizoum et al., 2011) were confined to the analysis of observations along the first cycle of loading–unloading, and only a few of them (Ayoub et al., 2011; Drozdov, 2011a,b) dealt with multi-cycle deformation. The novelty of our approach consists in the following: 1. Plastic strain εp is thought of as the sum of plastic strains in the crystalline εpc and amorphous εpa regions. Evolution of εpc and εpa with time t is governed by different flow rules. In particular, εpc increases under tension and decreases under retraction, while εpa grows monotonically under loading and unloading. 2. Strain energy density equals the sum of mechanical energies stored in individual chains and the energy of inter-chain interaction in the amorphous phase. The latter is treated as the quadratic function of plastic strain εpa with a coefficient R that evolves with number of cycles due to damage accumulation. 3. A stress–strain diagram is split into three groups of intervals: (i) stretching (strain ε increases and plastic strain εp exceeds maximum plastic strain εmax reached when the last retraction starts), p (ii) unloading (ε decreases), and (iii) reloading (ε increases, but εp < εmax ). p 4. Evolution of εpc (t) and εpa (t) under stretching, unloading, and reloading is described by the same differential equations, but with different coefficients. In our previous study (Drozdov and Christiansen, 2011), an attempt was undertaken (rather unsuccessful) to describe coincidence of stretching paths for virgin and cyclically preloaded specimens grounded on the first and last assumption only (viscoelastic effects and the energy of inter-chain interaction were disregarded, while transition to stretching was presumed to occur due to evolution of coefficients in the kinetic equation for εpc ascribed to damage accumulation under reloading). Although that model could reproduce feature (iii) of the Mullins effect, its applicability to the description of observations reported in the present study seems questionable because irreversible growth of damage is incompatible with fading memory of deformation history in semicrystalline polymers. The exposition is organized as follows. Observations on polypropylene in uniaxial tensile cyclic tests at room temperature are reported in Section 2. Constitutive equations for the viscoelastic and viscoplastic behavior of semicrystalline polymers are developed in Section 3. Adjustable parameters in the stress–strain relations are found in Section 4. Ability of the model to predict the fading memory phenomenon is confirmed in Section 5 by numerical simulation. In Section 6, fading memory of deformation history is demonstrated in other semicrystalline polymers (low density polyethylene under

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40.0

40.0

σ MPa

σ MPa A

0.0 40.0 σ MPa

C

0.0 40.0 σ MPa

B

0.0 0.0

0.16

D

0.0 0.0

0.16

Fig. 1. Stress  versus strain ε. Circles: experimental data in cyclic tests with N = 1 (A), N = 3 (B), N = 5 (C), N = 10 (D). Solid lines: results of numerical simulation.

cyclic tension with finite strains and poly(oxymethylene) under cyclic compression) and possible ways to extend the model to three-dimensional loading with large deformations are briefly discussed. Concluding remarks are formulated in Section 7. 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 Lyondell Basell (Rotterdam, Netherlands). Dumbbell specimens for mechanical tests (ASTM standard D-638) with cross-sectional area 10.0 mm × 3.8 mm were molded by using injection-molding machine Ferromatic Milacron K110. Uniaxial tensile tests were conducted at room temperature by means of universal testing machine Instron-5586 equipped with Instron 2630-113 static axial extensometer for control of longitudinal strains. Tensile force was measured by 50 kN load cell. The engineering stress  was determined as the ratio of axial force to cross-sectional area of undeformed specimens. The experimental program involved a series of cyclic tests and a relaxation test. Each test was carried out on a new specimen and repeated by twice. Observations reveal good reproducibility of measurements: deviations between stresses measured on different specimens do not exceed 3%. A series of tensile cyclic tests involved four experiments. In each test, a specimen was stretched with cross-head speed d˙ = 100 mm/min (corresponding to strain rate ε˙ = 2.0 × 10−2 s−1 ) up to the first maximum strain εmax , unloaded down to the mini1 mum stress  min = 1 MPa with the same strain cross-head speed, reloaded up to the maximum strain εmax , unloaded down to the 2 minimum stress  min , etc. Tests were conducted with N = 1, 3, 5, and 10 cycles. The ultimate strain reached in all tests equaled εmax = 0.15. Maximum strain in the nth cycle of a test with N cycles was calculated by the formula εmax = nεmax /N (n = 1, . . . , N). n The strain εmax = 0.15 was chosen to model the observations within the concept of small strains. The stress  min = 1 MPa was selected instead of  min = 0 to avoid buckling of specimens under retraction. The speed d˙ = 100 mm/min was chosen as a maximum cross-head speed with which the loading program was carried out with the required accuracy (maximum deviations between programmed and real values of strains εmax did not exceed 0.001). n Stress–strain diagrams in the cyclic tests are depicted in Fig. 1 where stress  is plotted versus strain ε. The following conclusions are drawn from this figure: (i) loading and unloading paths are strongly nonlinear and (ii) hysteresis energy (evaluated as the area between subsequent loading and unloading paths) increases noticeably with n. Comparison of the stress–strain curves is performed in Fig. 2 where two separate plots (for N = 1, 3 and N = 5, 10) are used to avoid overlapping of experimental data. Fig. 2 shows that not only loading paths of the stress–strain diagrams coincide when current

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40.0

Tensile strain ε is treated as the sum of elastic εe and plastic εp strains

σ MPa

ε = εe + εp . A

0.0 40.0

(1)

The plastic strain εp is split into the sum of two components εp = εpc + εpa

σ MPa B

0.0 0.0

0.16

Fig. 2. Stress  versus strain ε. Symbols: experimental data in cyclic tests with N = 1 (A – ), N = 3 (A – 䊉), N = 5 (B – ), and N = 10 (B – 䊉).

strain ε exceeds maximum strain in the previous cycle, but the corresponding unloading paths coincide as well. In other words, when ε slightly exceeds εmax , polypropylene looses its memory of n deformation history. To evaluate the time-dependent response, a tensile relaxation test was performed with strain ε = 0.03. First, a specimen was stretched with strain rate ε˙ = 2.0 × 10−2 s−1 up to the required strain. Afterwards, ε remained fixed, and a decrease in stress  was monitored as a function of time t. With reference to the ASTM protocol E-328, duration of short-term relaxation tests trel = 20 min was chosen. Experimental data in the relaxation test are reported in Fig. 3. Following common practice, the semi-logarithmic plot is employed, where stress  is depicted versus logarithm (log = log10 ) of relaxation time  = t − t0 (t0 stands for the instant when relaxation starts). The strain ε = 0.03 and strain rate ε˙ = 2.0 × 10−2 s−1 were selected from the condition that loading time did not exceed 2 s. 3. Constitutive modeling The mechanical response of a semicrystalline polymer under cyclic deformation is described within a two-step approach. At the first step, stress–strain relations are derived for an individual cycle of oscillations by means of the Clausius–Duhem inequality. At the other step, some material parameters in the constitutive equations (these quantities are treated as constants along each cycle of oscillations) are allowed to change with number of cycles. Kinetic equations for their evolution reflect the effect of damage accumulation on the viscoplastic response. 3.1. Stress–strain relations Stress–strain relations in cyclic viscoelastoplasticity of semicrystalline polymers under three-dimensional deformation with small strains have been developed in Drozdov (2011a,b). A version of these equations is presented for uniaxial tensile deformation with an arbitrary program ε(t).

(2)

that reflect inelastic deformations in the crystalline and amorphous phases. The strain rate for plastic deformation in the crystalline phase is proportional to strain rate for macro-deformation dεpc dε = . dt dt

(3)

The non-negative function  (i) vanishes in the reference state (which means that no plastic deformation occurs at very small strains), (ii) monotonically grows under tension and decreases under retraction (which reflects stress-induced acceleration of plastic flow), and (iii) reaches its ultimate value ∞ = 1 at relatively large strains, when the rates of plastic strain and macro-strain coincide. Under stretching, evolution of  with time obeys the equation d 2 dε = a(1 − ) dt dt

(4)

with a constant coefficient a = a1 > 0. Under unloading and reloading,  remains constant (which means that corresponding coefficients a2 and a3 in Eq. (4) vanish). The strain rate for plastic deformation in the amorphous matrix is governed by the equation



dεpa = S εe (t) − Rεpa (t) − dt



0

   dε  f (v)Z(t, v)dv   , dt

(5)

where R and S adopt non-negative values R1 , R2 , R3 , and S1 , S2 , S3 under stretching, unloading, and reloading. The integral term in Eq. (5) describes the effect of viscoelasticity on plastic flow in the amorphous phase: dumb variable  characterizes dimensionless activation energy for rearrangement of chains, the function f (v) accounts for inhomogeneity of the amorphous phase caused by the presence of spherulites, and Z(t, v) reflects a decay in stress induced by relaxation. The function f (v) is given by



f (v) = f0 exp −

1 2

 v 2 

(v ≥ 0),

˙

f (v) = 0 (v < 0),

(6)

where ˙ > 0 characterizes distribution of relaxation  ∞ times, and f0 is determined from the normalization condition 0 f (v)dv = 1. The function Z(t, v) satisfies the linear equation ∂Z (t, v) =  (v)[εe (t) − Z(t, v)], ∂t

Z(0, v) = 0

(7)

with  (v) = exp(−v), where stands for relaxation rate. The stress  reads





(t) = E[1 − (t)] εe (t) − 30.0

∞



∞

f (v)Z(t, v)dv ,

(8)

0

where E denotes Young’s modulus. σ MPa

3.2. Evolution of material parameters

0.0 0.0

logτ s

3.5

Fig. 3. Stress  versus relaxation time . Circles: experimental data in relaxation test. Solid line: results of numerical simulation.

Eqs. (1)–(8) provide stress–strain relations for uniaxial tension of a specimen with an arbitrary program ε(t). These equations involve with 10 adjustable parameters: (i) E stands for elastic modulus, (ii) denotes relaxation rate, (iii) ˙ determines distribution of relaxation times, (iv) a1 characterizes growth of rate of inelastic deformation in the crystalline phase under stretching, (v) S1 , S2 , S3 stand for rates of viscoplastic deformation in the amorphous phase under stretching, unloading, and reloading, and (vi) R1 , R2 ,

A.D. Drozdov et al. / Mechanics Research Communications 48 (2013) 70–75

R3 account for changes in energy of interaction between chains in the amorphous matrix at unloading and reloading. As the number of adjustable parameters is not small, the following scenario is proposed to describe the effect of damage accumulation on their evolution: 1. Parameters E, , and ˙ are constants. The statement that elastic modulus E remains constant distinguishes the present approach from conventional concepts in damage mechanics that postulate a decay in modulus under deformation. The assertion that and ˙ are constants (which means that plastic flow in a semicrystalline polymer does not affect its viscoelastic response) is confirmed by experimental data reported in Drozdov and Dusunceli (2012). 2. Coefficients a1 and S1 (that characterize the mechanical response of a semicrystalline polymer under stretching) are constants. Damage-induced changes in coefficient R1 are described by the first-order kinetic equation dR1 = ˛1 (R1∞ − R1 ), dWp

(9)

where ˛1 , R1∞ are constants, and



t

Wp (t) =

(s) 0

dεp (s)ds dt

(10)

stands for plastic work per unit volume. 3. The mechanical behavior along each retraction path is described by two parameters, R2 and S2 . Evolution of these quantities driven by damage accumulation is governed by the kinetic equations dR2 = ˛2 (R2∞ − R2 ), dWp

dS2 = ˇ2 (S2∞ − S2 ), dWp

(11)

where ˛2 , R2∞ , ˇ2 , S2∞ are constants. 4. The response along each reloading path is determined by two coefficients, R3 and S3 . The parameter S3 obeys the kinetic equation dS3 = ˇ3 (S3∞ − S3 ), dWp

(12)

where ˇ3 , S3∞ are constants. The effect of specific plastic work on coefficient R3 is non-monotonic: R3 decreases linearly with Wp at the initial stage of damage accumulation, reaches its minimum value, and grows afterwards, R3 = R30 − R31 Wp

(Wp < Wp0 ),

R3 = R32 + R33 Wp

(Wp > Wp0 ), (13)

where

R30 ,

R31 ,

R32 ,

R33

are constants.

Our aim now is to provide some remarks regarding Eqs. (9) and (11)–(13). 1. Unlike conventional concepts in damage mechanics, no internal variables are introduced to characterize damage accumulation in semicrystalline polymers: evolution of internal structure is governed by specific plastic work Wp only. This assertion is rather strong: it means that parameters R1 , R2 , R3 and S2 , S3 determined by matching observations in cyclic tests with various deformation programs (each set of experimental data is fitted separately) lie on the same curves when these quantities are plotted versus Wp . 2. Eqs. (11) and (12) state that coefficients S2 and S3 (these parameters characterize rates of plastic flow in the amorphous matrix under retraction and reloading, respectively) decrease exponentially with Wp and approach their ultimate values, S2∞ and

73

Table 1 Adjustable parameters in the constitutive equations. Parameter

Value

E (GPa) (s–1 ) ˙ a1 S1

2.48 0.33 12.0 13.1 49.4

S3∞ at large Wp . When plastic flow in the amorphous phase is treated as sliding of junctions between polymer chains with respect to their reference positions, its slowing down induced by damage accumulation may be explained by two reasons: (i) nucleation, growth, and coalescence of micro-voids and microcracks (Galeski and Rozanski, 2010; Boisot et al., 2011) that prevent sliding in close surroundings of the defects, and (ii) fragmentation of weak lamellae and slippage of broken lamellar parts into the amorphous matrix (Stribeck et al., 2008; Drozdov, 2011c) where they serve as obstacles for sliding of junctions. 3. The exponential decay in coefficient R2 [Eq. (11)] and the linear decrease in R3 [Eq. (13)] at the initial period of damage accumulation are attributed to formation of micro-defects in the amorphous matrix which reduce the energy of inter-chain interaction. 4. An increase in coefficients R1 [Eq. (9)] and R3 [Eq. (13)] are associated with the growth of interaction energy due to orientation of broken lamellar fragments along the loading direction. 4. Adjustable parameters Adjustable parameters in the stress–strain relations are found by fitting the experimental data depicted in Figs. 1 and 3. Each set of observations is approximated separately. First, coefficients and ˙ are determined by matching the relaxation curve in Fig. 3. Afterwards, parameters E, a1 , and S1 are found by fitting loading path of the stress–strain curve with N = 1. Finally, coefficients R1 , R2 , R3 , S2 , S3 are calculated that ensure the best fit (for each cycle of oscillations) of the stress–strain diagrams in Fig. 1. Integration of constitutive equations (1)–(8) over time t is performed by the Runge–Kutta method with step t = 0.01 s. Evaluation of the integral in Eq. (8) is conducted by the Simpson method with v = m v,

v = 2.0, and m = 0, 1, . . . , 20. The best-fit values of E, , , a1 , S1 are listed in Table 1. Parameters R1 , R2 , R3 , S2 , S3 are plotted versus plastic work Wp in Fig. 4 together with their fits by Eqs. (9) and (11)–(13). This figure shows that Eqs. (9) and (11)–(13) ensure an acceptable approximation of the experimental dependencies. Results of numerical simulation are presented in Figs. 1 and 3 together with observations in cyclic tests and relaxation test. These 1.2 R1 ∗ 1.2 8.0 R2 0.0 1.1 ∗ R3

20.0

∗

∗

0.0 50.0

∗

∗

∗

∗ Wp MPa

6.0

∗

∗

B

∗

S3 A

0.7 0.0

∗

S2

∗

0.0 0.0

Wp MPa

6.0

Fig. 4. Parameters R1 , R2 , R3 (A) and S2 , S3 (B) versus plastic work Wp . Symbols: treatment of observations in cyclic tests ( – N = 10; 䊉 – N = 5; ∗ – N = 3; – N = 1). Solid lines: approximation of the data by Eqs. (9) and (11)–(13) with ˛1 = 0.98, R1∞ = 1.13, ˛2 = 1.63, R2∞ = 0.055, R30 = 1.01, R31 = 0.080, R32 = 0.83, R33 = 0.036, ˇ2 = 3.09, S2∞ = 8.77, ˇ3 = 0.71, and S3∞ = 11.13.

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A.D. Drozdov et al. / Mechanics Research Communications 48 (2013) 70–75

40.0

40.0

σ MPa

σ MPa

A

0.0 0.0

0.16

B

0.0 0.0

0.16

Fig. 5. Stress  versus strain ε. A – symbols: results of numerical simulation for cyclic tests with N = 3, εmax = 0.06, 0.12, 0.16 () and N = 4, εmax = 0.04, 0.08, 0.12, 0.16 ( ). B – circles: experimental data in the cyclic test with N = 1. Solid line: results of numerical simulation for cyclic test with N = 7, εmax = 0.03, 0.05, 0.07, 0.09, 0.11, 0.13, and 0.15.

figures demonstrate good agreement between the experimental data and their description by the model. 5. Numerical simulation To evaluate ability of the constitutive equations to predict observations in cyclic tests, numerical integration of Eqs. (1)–(13) is conducted with the material constants listed in Table 1. Results of numerical analysis for cyclic tests with strain rate ε˙ = 0.02 s−1 , minimum stress  min = 1 MPa, and various numbers of cycles N and (n = 1, . . . , N) are reported in Fig. 5. maximum strains εmax n In Fig. 5A, two stress–strain diagrams are presented for multi= 0.06, 0.12, 0.16, and cycle tensile deformations with N = 3, εmax n N = 4, εmax = 0.04, 0.08, 0.12, 0.16. According to this figure, the n stress–strain curves along the last two cycles (with the same maximum strains) entirely coincide despite the fact that the deformation programs differ noticeably along previous cycles. This confirms ability of the model to predict fading memory of deformation history in semicrystalline polymers. In Fig. 5B, the stress–strain diagram is plotted for cyclic deforma= 0.01 + 0.02n together with observations in tion with N = 7, εmax n the test with N = 1, εmax = 0.15. This figure demonstrates that at 1 each cycle of oscillations, stretching paths of the stress–strain curve for multi-step cyclic deformation reach that of the experimental diagram for one-cycle program, and the corresponding retraction paths coincide. 6. Discussion The aim of this section is twofold: (i) to report experimental data on other semicrystalline polymers that reveal the fading memory phenomenon in multi-cycle tests under various loading conditions (tensile tests with large deformations and compressive tests with small strains), and (ii) to suggest possible ways for extension of the stress–strain relations to cyclic three-dimensional deformations with finite strains. We begin with observations in multi-cycle tensile tests with finite strains on low density polyethylene (LDPE) Riblene FL 20 (density 0.92 g/cm3 , melt flow rate 2.2 g/10 min, melting temperature 109 ◦ C) supplied by Polimeri Europa (Italy). Mechanical tests were performed at room temperature on injection-molded dumbbell specimens with the same dimensions as the polypropylene samples. The experimental program involved four tests carried out with cross-head speed d˙ = 100 mm/min (that corresponded to strain rate ε˙ = 2.0 × 10−2 s−1 ). In the first three tests, specimens

= 0.1, 0.3, and 0.5, were stretched up to maximum strains εmax 1 unloaded down to minimum stress  min = 1 MPa, and reloaded up to εmax = εmax . In the fourth test, a specimen was stretched up to 2 1 max ε1 = 0.1, retracted down to  min = 1 MPa, reloaded up to εmax = 2 0.3, retracted down to  min = 1 MPa, reloaded up to εmax = 0.5, 3 retracted down to  min = 1 MPa, and reloaded up to εmax = εmax . 4 3 Experimental data are presented in Fig. 6 where engineering stress  is plotted versus engineering strain ε. We proceed with multi-cycle compressive tests on poly (oxymethylene) copolymer (POM-C SAN) Riatal (density 1.41 g/cm3 , melting temperature 165 ◦ C) supplied by RIAS A/S (Denmark) in the form of cylindrical rods. Mechanical tests were performed at room temperature on cylindrical specimens with diameter 20.9 mm and height 48.5 mm machined from the rods. The experimental program involved three tests conducted with cross-head speed d˙ = 5 mm/min (that corresponded to strain rate ε˙ = 1.7 × 10−3 s−1 ). In each test, a specimen was compressed down to maximum compressive strain εmax , unloaded up to minimum 1 stress  min = 1 MPa, compressed down to strain εmax , unloaded 2 up to minimum stress  min = 1 MPa, etc. Maximum compressive strains were chosen according to the rule εmax = 0.2 n/N. Observan tions in cyclic compressive tests with N = 2, 4, and 5 are reported in Fig. 6 where compressive stress  is plotted versus compressive strain ε. According to Figs. 6 and 7, the fading memory phenomenon (coincidence of unloading and reloading paths of the stress–strain diagrams of specimens subjected to different loading histories) is observed not only at small tensile deformations, but also in tests with large tensile deformations and under compression. Due to the fact that the structure of constitutive equations (1)–(8) is independent of sign of strain ε, these relations can 12.0 σ MPa

0.0 0.0

0.5

Fig. 6. Stress  versus strain ε. Circles: experimental data on LDPE in multi-cycle = 0.1, 0.3, 0.5 and  min = 1 MPa. Other symbols: experimental data test with εmax n = 0.1 ( ), εmax = 0.3 (∗), εmax = 0.5 (䊉), and  min = 1 MPa. in cyclic tests with εmax 1 1 1

A.D. Drozdov et al. / Mechanics Research Communications 48 (2013) 70–75

100.0 σ MPa

0.0 0.0

0.2

Fig. 7. Stress  versus strain ε. Symbols: experimental data on POM in multi-cycle = 0.2 n/N and  min = 1 MPa ( – N = 5; 䊉 – N = 4; and compressive tests with εmax n – N = 2).

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deformations. Material constants in the stress–strain relations are found by fitting the observations. Numerical analysis reveals that the effect of damage accumulation on the mechanical response can be accounted for by means of specific plastic work. Results of simulation demonstrate ability of the model to describe the mechanical response of polypropylene in multi-step cyclic tests and to predict the fading memory effect. Further analysis of this phenomenon requires investigation of the effects of temperature (observations are reported at room temperature only), strain rate (only cyclic unloading with a constant cross-head speed is used), and deformation mode (transition from uniaxial tension to equi-biaxial tension and shear). References

be applied to the analysis of multi-cycle compressive loadings without changes. To describe mechanical response under threedimensional deformations with finite strains, Eq. (1) should be transformed into the conventional multiplicative decomposition of the deformation gradient F, while Eq. (2) can be replaced with Dp = Dpc + Dpa , where Dp , Dpa , Dpc stand for the rate-of-strain tensors for corresponding plastic deformations defined in the actual configuration. The finite-deformation analog of Eq. (3) reads Dpc = D, where D denotes the rate-of-strain tensor for macro-deformation. Derivation of counterparts of Eqs. (5) and (8) is straightforward because they follow from the Clausius–Duhem inequality where strain energy density W is presented as the sum of two components (mechanical energy stored in polymer chains W1 and energy of inter-chain interaction W2 that depend on principal invariants of the Cauchy–Green tensors for elastic Be and plastic Bpa deformations, respectively). The main difficulty in the analysis of multi-cycle three-dimensional deformations with finite strains consists in distinguishing between loading and unloading paths under non-proportional loading (which is typical of all models in finite viscoplasticity) and discriminating reloading and stretching paths (which is characteristic of the present approach). It appears max , where J that a counterpart (for example, J1p = J1p 1p stands for the first principal invariant of the Cauchy–Green tensor Bp ) may serve , but this hypothas an extension of the proposed criterion εp = εmax p esis requires experimental validation that is beyond the scope of this study. 7. Concluding remarks Observations are reported on isotactic polypropylene in multicycle uniaxial tensile tests with increasing maximum strains at room temperature that reveal fading memory of deformation history: when maximum strains εmax coincide for two deformation n programs with different εmax (k = 1, . . . , n − 1), the corresponding k stress–strain diagrams coincide as well. The fact that this phenomenon is rather typical of semicrystalline polymers is confirmed by experimental data in cyclic tests on low density polyethylene (tensile deformations with finite strains) and poly(oxymethylene) (compressive loadings). Constitutive equations are developed in cyclic viscoelasticity and viscoplasticity of semicrystalline polymers with small strains and some clues are provided for their extension to finite

Avanzini, A., 2008. Mechanical characterization and finite element modelling of cyclic stress–strain behaviour of ultra high molecular weight polyethylene. Materials and Design 29, 330–343. Ayoub, G., Zairi, F., Nait-Abdelaziz, M., Gloaguen, J.M., 2010. Modelling large deformation behaviour under loading–unloading of semicrystalline polymers: application to a high density polyethylene. International Journal of Plasticity 26, 329–347. Ayoub, G., Zairi, F., Nait-Abdelaziz, M., Gloaguen, J.M., 2011. Modeling the low-cycle fatigue behavior of visco-hyperelastic elastomeric materials using a new network alteration theory: application to styrene-butadiene rubber. Journal of the Mechanics and Physics of Solids 59, 473–495. Boisot, G., Laiarinandrasana, L., Besson, J., Fond, C., Hochstetter, G., 2011. Experimental investigation and modeling of volume change induced by void growth in polyamide 11. International Journal of Solids and Structures 48, 2642–2654. Diani, J., Fayolle, B., Gilormini, P., 2009. A review on the Mullins effect. European Polymer Journal 45, 601–612. Drozdov, A.D., 2011a. Multi-cycle viscoplastic deformation of polypropylene. Computation Materials Science 50, 1991–2000. Drozdov, A.D., 2011b. Cyclic viscoelastoplasticity and low-cycle fatigue of polymer composites. International Journal of Solids and Structures 48, 2026–2040. Drozdov, A.D., 2011c. Cyclic strengthening of polypropylene under strain-controlled loading. Materials Science and Engineering A 528, 8781–8789. Drozdov, A.D., Christiansen, J.deC., 2008. Cyclic elastoplasticity of solid polymers. Computation Materials Science 42, 27–35. Drozdov, A.D., Christiansen, J.deC., 2011. Mullins’ effect in semicrystalline polymers: experiments and modeling. Meccanica 46, 359–370. Drozdov, A.D., Dusunceli, N., 2012. Universal mechanical response of polypropylene under cyclic deformation. Journal of Polymer Engineering 32, 31–42. Dusunceli, N., Colak, O.U., 2008. Modelling effects of degree of crystallinity on mechanical behavior of semicrystalline polymers. International Journal of Plasticity 24, 1224–1242. Galeski, A., Rozanski, A., 2010. Cavitation during drawing of crystalline polymers. Macromolecular Symposium 298, 1–9. Hassan, T., Colak, O.U., Clayton, P.M., 2011. Uniaxial strain and stress-controlled cyclic responses of ultra high molecular weight polyethylene: experiments and model simulations. Transactions of the ASME – Journal of Engineering Materials and Technology 133, 021010. Hizoum, K., Remond, Y., Patlazhan, S., 2011. Coupling of nanocavitation with cyclic deformation behavior of high-density polyethylene below the yield point. Transactions of the ASME – Journal of Engineering Materials and Technology 133, 030901. Mizuno, M., Sanomura, Y., 2009. Phenomenological formulation of viscoplastic constitutive equation for polyethylene by taking into account strain recovery during unloading. Acta Mechanica 207, 83–93. Mullins, L., 1947. Effect of stretching on the properties of rubber. Journal of Rubber Research 16, 275–289. Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society A 455, 2861–2877. Silberstein, M.N., Boyce, M.C., 2010. Constitutive modeling of the rate, temperature, and hydration dependent deformation response of Nafion to monotonic and cyclic loading. Journal of Power Sources 195, 5692–5706. Stribeck, N., Nochel, U., Funari, S., Schubert, T., Timmann, A., 2008. Nanostructure evolution in poly(propylene) during mechanical testing. Macromolecular Chemistry and Physics 209, 1999–2002.