Essential work of fracture and viscoplastic response of a carbon black-filled thermoplastic elastomer

Engineering Fracture Mechanics 76 (2009) 1977–1995

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Essential work of fracture and viscoplastic response of a carbon black-filled thermoplastic elastomer A.D. Drozdov *, S. Clyens, J. Christiansen Danish Technological Institute, Gregersensvej 1, DK-2630 Taastrup, Denmark

a r t i c l e

i n f o

Article history: Received 12 August 2008 Received in revised form 17 March 2009 Accepted 10 May 2009 Available online 18 May 2009 Keywords: Polymer matrix composites Constitutive modelling EWF concept Viscoplasticity Strain rate effects

a b s t r a c t Observations are reported on a carbon black-filled thermoplastic elastomer in uniaxial cyclic tensile tests with various maximum strains and double-edge-notched-tensile (DENT) tests with various ligament widths at ambient temperature. It is shown that the stress– strain diagrams in DENT tests measured relatively far away from the ligament coincide with those in tensile cyclic tests on un-notched samples. To describe the viscoplastic response of un-notched specimens, constitutive equations are derived, and adjustable parameters are found by fitting the experimental data. It is demonstrated how the energy stored in a DENT sample under tension can be accounted for in calculations of the specific essential work of fracture. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction This paper is concerned with experimental evaluation of the essential work of fracture and modeling the viscoplastic behavior of carbon black-reinforced thermoplastic elastomers (TPE). Experiments are performed on Thermoplast K, a hydrogenated styrene block copolymer (HSBC)-based thermoplastic elastomer reinforced with about 70 wt.% of carbon black particles. As this composite demonstrates strong thermal and chemical resistance, it can be employed as a sealing material for low-temperature proton exchange membrane fuel cells (PEM FCs) with operating temperatures up to 100 °C. One of the main characteristics of a seal is its life-time, that is duration of exposure at an operative temperature at which no substantial deterioration is observed of mechanical properties. Our preliminary study shows a substantial decrease in elongation to break of Thermoplast K with exposure time (when duration is of the order of hours), which implies that toughness of this material is strongly affected by thermal degradation. As elongation to break cannot be employed as a direct measure of fracture (this quantity strongly depends on deformation mode and strain rate), it seems reasonable to apply the concept of essential work of fracture (EWF) to assess the life-time of sealing materials. The EWF concept has been developed several decades ago [1–3]. Its main advantage is that the material parameter under investigation (the specific essential work of fracture we ) is determined by applying a rather simple procedure to observations on double-end-notched (DENT) samples in uniaxial tensile tests. A specimen (Fig. 1) is a thin rectangular plate (with length l, width h, and thickness b), where two symmetric straight cuts (with length r) are made separated by a ligament of width H (with H ¼ h  2r). In a DENT test, the plate is stretched with a constant cross-head speed d_ in the direction of its length (perpendicular to the cuts) and the tensile force f is measured as a function of displacement d up

* Corresponding author. Tel.: +45 72 20 31 42; fax: +45 72 20 31 12. E-mail address: [email protected] (A.D. Drozdov). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.05.003

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Nomenclature a coefficient in Eq. (19) ~ a coefficient in Eq. (21) b thickness of a sample c coefficient in Eq. (5) d tensile displacement d_ cross-head speed D strain rate intensity F function defined by Eq. (27) f tensile load h width of a sample H ligament width J number of subintervals first principal invariant of the Cauchy–Green tensor for elastic deformation J e1 first principal invariant of the relative Cauchy–Green tensor for elastic deformation je1 k elongation ratio maximum elongation ratio kmax elongation ratio for plastic deformation kp l length of a sample i; j; m indices p pressure Q rate of internal dissipation r length of a cut T temperature t time length of time interval, where elastic modulus changes monotonically t0 W work of fracture w specific work of fracture essential work of fracture We specific essential work of fracture we we0 ; we1 coefficients in Eq. (7) strain energy density of an equivalent network W eq hysteresis energy W hys W hys0 ; W hys1 coefficients in Eq. (29) specific hysteresis energy whys work of plastic deformation Wp specific work of plastic deformation wp X 1 ; X 2 ; X 3 Cartesian coordinates in the reference state x1 ; x2 ; x3 Cartesian coordinates in the actual state left Cauchy–Green tensor for elastic deformation Be relative left Cauchy–Green tensor for elastic deformation be right Cauchy–Green tensor for elastic deformation Ce D rate-of-strain tensor for macro-deformation rate-of-strain tensor for elastic deformation De rate-of-strain tensor for plastic deformation Dp F deformation gradient for macro-deformation deformation gradient for elastic deformation Fe deformation gradient for plastic deformation Fp I unit tensor L rate-of-strain tensor for macro-deformation rate-of-strain tensor for elastic deformation Le rate-of-strain tensor for plastic deformation lp S tensor defined by Eq. (A.1) T extra stress tensor W vorticity tensor for macro-deformation vorticity tensor for elastic deformation We a coefficient in Eq. (20) a~ coefficient in Eq. (21) a0 ; a1 coefficients in Eq. (23) b coefficient in Eq. (20)

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~ b coefficient in Eq. (21) b0 ; b1 coefficients in Eq. (23) d dimensionless parameter of a plastic zone Dk step of integration Da; Da; Db; DU lengths of subintervals  tensile strain e elastic tensile strain max maximum strain max 0 ; max 1 coefficients in Eq. (4) _ tensile strain rate _ A ; _ B strain rates measured by an extensometer in positions A and B l elastic modulus l0 ; l1 coefficients in Eq. (22) n1 ; n2 ; n3 Cartesian coordinates in the intermediate configuration qdef ; qundef density of deformed and undeformed samples r engineering tensile stress rexp tensile stress measured in a test rnum tensile stress calculated in simulation s time U coefficient in Eq. (19) / function characterizing plastic flow in Eq. (11) w coefficient in Eq. (18) R Cauchy stress tensor

to breakage of specimen. Given an experimental load–displacement diagram f ðdÞ, the work of fracture W is calculated as area under the curve f ðdÞ. To interpret the experimental data, it is assumed that the plate can be split into three regions (the ligament, a plastic zone surrounding the ligament, and the remaining part of the sample). Under loading, the work of tensile forces is transformed into the energy of breakage of the ligament (the essential work of fracture W e ) and the energy of plastic deformation (the energy stored in the plastic zone W p ). The remaining part of the sample is treated as a purely elastic medium that returns the entire stored energy at breakage of the ligament. The following assumptions are formulated: (i) W e is proportional to cross-sectional area bH of the ligament, (ii) W p is proportional to volume of 2 the plastic zone, and (iii) volume of the plastic zone reads dbH , where d 2 ð0; 1Þ is a constant. Equating the work of fracture W to the sum of energy of fracture of the ligament W e and work of plastic deformation W p and dividing the result by bH, we find that

w ¼ we þ dwp H;

ð1Þ

where

Fig. 1. Positions A and B of extensometer in DENT tests.

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w¼

W ; bH

we ¼

We ; bH

wp ¼

Wp dbH

2

:

ð2Þ

The specific essential work of fracture we and the coefficient dwp are determined by fitting the experimental curve wðHÞ by linear dependence (1). The EWF theory can be applied to the analysis of experimental data on DENT samples with arbitrary thicknesses b. An additional benefit is that for relatively thin specimens (when plane-stress conditions are satisfied), we is expressed in terms of stress intensity factors by means of the J-integral technique [4–6]. In the past two decades, the EWF concept has been applied to evaluate the specific essential work of fracture for a number of neat and particle-reinforced amorphous and semicrystalline polymers and their blends (with typical values of we belonging to the interval between 10 and 60 kJ=m2 ). Under fixed experimental conditions (cross-head speed and temperature) and dimensions of samples, physically plausible results are obtained when the ligament width H remains relatively large ðH > 0:7hÞ [7] (see [8,9] for more sophisticated estimates). Even under this restriction (that results in a rather low accuracy of determining we ), some modifications of the standard algorithm are needed when extensive necking of ligament is observed [10,11]. At small ligament widths ðH < 0:7hÞ, Eq. (1) may be violated, and basic assumptions of the EWF concept need to be reformulated [12,13] to account for a nonlinearity of the dependence w(H). Experimental investigation of the effect of temperature T on specific essential work of fracture shows that we is either independent of T [14–16] or monotonically decreases with temperature [17,18]. The influence of cross-head speed d_ on we appears to be more sophisticated. For some polymers, the specific essential work of fracture is independent of strain rate [20], while for others, we increases with cross-head speed [21,22], decreases with it [13], or demonstrates a non-monotonic dependence on d_ [23–25]. Analysis of the effect of parameters l, h, and b (that characterize geometry of specimens) on specific essential work of fracture leads to contradictory results: some authors show that we is independent of these quantities (or, at least, some of them) [8,19,20,26–28], while the others reveal a strong effect of sample geometry on we [29,30]. This brief survey demonstrates that the EWF concept serves as an important tool for evaluation of toughness of polymers and polymer composites, but results of the conventional analysis of observations should be treated with caution. As one of the reasons for the latter conclusion, modeling of the main part of a sample (outside a plastic zone surrounding the ligament) as an elastic continuum may be mentioned. As volume of this part of a specimen substantially exceeds that of the domain where noticeable plastic deformations (formation, growth, and coalescence of micro-voids and micro-cracks) are observed after breakage of the ligament, the entire energy of irreversible deformation outside the plastic zone (the hysteresis energy calculated as area between the stress–strain diagrams under loading and subsequent unloading) can be comparable (or even exceed) the energy W p , which leads to violation of Eq. (1). The objective of this study is three-fold: 1. To report experimental data on Thermoplast K (DENT samples with various ligament lengths H) when conventional load– displacement diagrams are observed simultaneously with stress–strain curves outside the plastic zone (importance of measurement of strains in DENT tests is discussed in [31]). 2. To calculate the specific essential work of fracture we by using (i) the standard approach and (ii) a modified concept (that accounts for the energy stored in the main part of a DENT sample) and to demonstrate that these quantities practically coincide at room temperature. 3. To derive constitutive equations for the viscoplastic response of a polymer composite at cyclic deformations with finite strains and to show that the model allows the specific essential work of fracture to be determined by using observations in conventional DENT tests only. The exposition is organized as follows: observations in DENT tests and uniaxial cyclic tensile tests are reported in Section 2. In Section 3, the specific essential work of fracture is calculated by using the standard approach, as well as by applying a modified EWF concept. A constitutive model in finite viscoplasticity of polymer composites is developed in Section 4. Adjustable parameters in the stress–strain relations are found in Section 5 by fitting the experimental data. Concluding remarks are formulated in Section 6. Thermodynamic consistency of the constitutive equations is discussed in Appendix A.

2. Experimental results Carbon black-reinforced thermoplastic elastomer Thermoplast K TV5LVZ [density 1:07 g=cm3 , melt flow index 12 g/ 10 min at 230 °C, elongation at break 520%, hardness (shore A) 50] was purchased from Kraiburg TPE GmgH (Germany). Dumbbell specimens for uniaxial tensile tests (ASTM standard D638) with cross-sectional area 9.7 mm  4.0 mm and length of the active zone 100 mm were molded by using injection-molding machine Arburg 320C. Samples for DENT tests were prepared by making symmetric cuts in the middle of these specimens by means of a saw with the disk thickness 0.6 mm. Pre-notching was followed by sharpening with a razor blade to get the lengths r = 1.5, 2.0, 2.5, 2.8, and 4.2 mm that were controlled under an optical microscope.

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Tensile tests were conducted with the help of universal testing machine Instron-5569 equipped with an electro-mechanical sensor for control of longitudinal strains in the active zone of samples. The tensile force was measured by a standard load cell. The first series of experiments involved 5 DENT tests. In each test, a specimen was stretched up to breakage with a constant cross-head speed d_ ¼ 100 mm=min. The tests were repeated a minimum of twice, first, when the extensometer was in position A, and, afterwards, in position B (Fig. 1). The experimental data are reported in Fig. 2, where the tensile load f is plotted versus displacement d. According to this figure, (i) the load f and the maximum extension of sample strongly increase with H, and (ii) neck propagation is not observed (this phenomenon is characterized by pronounced changes in slope of the falling part of a load–displacement diagram [10,11]). Dependencies of engineering tensile strain on time were measured at two positions of the extensometer. Typical experimental data (for samples with H = 5.7 mm) are presented in Fig. 3, where the strain  is plotted versus time t. Observations are approximated by the linear relation

 ¼ _ t;

ð3Þ

where the strain rate _ is found by the least-squares algorithm. Fig. 3 shows that the DENT tests were conducted with practically constant strain rates. The strain rate measured by the extensometer in position B, _ B ¼ 1:4  102 s1 , is very close to

Fig. 2. Tensile load f versus displacement d. Symbols: experimental data in DENT tests with the cross-head speed d_ ¼ 100 mm= min on samples with various ligament widths H mm.

Fig. 3. The engineering strain  versus time t. Symbols: experimental data on samples with H = 5.7 mm in DENT tests with the cross-head speed d_ ¼ 100 mm= min for two positions of extensometer. Solid lines: their approximation by Eq. (3).

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that corresponding to uniaxial tensile deformation of an un-notched sample with the same cross-head speed. The strain rate measured by the extensometer in position A, _ A ¼ 2:0  102 s1 , exceeds _ B by 43%. The stress–strain diagrams for the main part of a specimen (sufficiently far away from the ligament) are presented in Fig. 4 (triangles), where the engineering stress r (ratio of the axial force to the cross-sectional area of an undeformed specimen) is plotted versus tensile strain  (measured by the extensometer in position B). Given a ligament width H, the stress r increases with , reaches its maximum at the instant when breakage of the ligament starts (the tensile strain at this point is denoted as max ), and decreases afterwards. The dependence of maximum strain max on ligament width H is reported in Fig. 5. This figure shows good agreement between the experimental data and their approximation by the linear equation

max ¼ max0 þ max1 H;

ð4Þ

where the coefficients max0 and max1 are calculated by the least-squares method. The other series of experiments involved five uniaxial tensile loading–unloading tests on un-notched specimens with the cross-head speed d_ ¼ 100 mm=min and the maximum strains max coinciding with those measured in DENT tests. The experimental stress–strain diagrams are depicted in Fig. 4. This figure demonstrates excellent agreement between the stress– strain curves observed in DENT tests and cyclic tensile tests.

Fig. 4. Engineering stress r versus tensile strain . Triangles: experimental data in DENT tests with the cross-head speed d_ ¼ 100 mm= min. Other symbols: experimental data in cyclic tensile tests with the same cross-head speed on un-notched samples. Solid lines: results of numerical simulation.

Fig. 5. Maximum tensile strain max versus ligament width H. Circles: experimental data in DENT tests with the cross-head speed 100 mm/min. Solid line: their approximation by Eq. (4) with max 0 ¼ 0:10 and max 1 ¼ 0:12 mm1 .

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Two arguments may be applied to explain coincidence of the stress–strain diagrams measured on notched and unnotched specimens. Concurrence of the loading paths of these diagrams (when stretching of samples occurs with the same strain rate) may be thought of as a consequence of Saint–Venant’s principle [32–34], according to which, distribution of stresses far away from the boundary of a deformed body is weakly affected by boundary conditions (perturbations of external load driven by the presence of ligament). Closeness of the unloading paths of the stress–strain curves (when specimens are deformed with different strain rates) is attributed to the fact that the mechanical response of Thermoplast K at unloading is weakly affected by strain rate [36]. For each ligament width H, a stress–strain diagram was measured on an un-notched specimen in a cyclic test with the maximum strain max coinciding with the maximum strain observed in an appropriate DENT test. This stress–strain dependence was recalculated to give an appropriate load–displacement curve (these curves are not presented as they differ from those depicted in Fig. 4 in the scales of horizontal and vertical axes only). The hysteresis energy W hys was found as the area between the loading and unloading paths of this load–displacement diagram. The dependence of the specific hysteresis energy whys ¼ W hys =ðbHÞ on H is plotted in Fig. 6. The experimental data are approximated by the linear equation

whys ¼ cH;

ð5Þ

Fig. 6. Specific hysteresis energy whys versus ligament length H. Circles: treatment of experimental data in cyclic tensile tests on un-notched samples. Solid line: their approximation by Eq. (5) with c ¼ 9:0  103 J=mm3 .

Fig. 7. Tensile load f versus displacement d. Symbols: experimental data in DENT tests with various cross-head speeds d_ mm/min on samples with the ligament width H ¼ 4:7 mm.

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Fig. 8. Engineering stress r versus tensile strain . Triangles: experimental data in DENT tests with various cross-head speeds d_ mm= min on samples with the ligament width H ¼ 4:7 mm. Other symbols: experimental data in cyclic tensile tests with the same cross-head speeds on un-notched samples. Solid lines: results of numerical simulation.

where c is found by the least-squares technique. Fig. 6 shows that Eq. (5) (which implies that the specific hysteresis energy is proportional to ligament width H) correctly describes the observations. The last series of experiments consisted of three tensile tests on DENT samples with H = 4.7 mm conducted with the cross-head speeds d_ ¼ 1, 10, and 100 mm/min (with the extensometer in position B) and appropriate cyclic tensile tests with the same cross-head speeds and the maximum strains max that coincided with the maximum strains measured in DENT tests. The cross-head speeds in our experiments covered the conventional interval of cross-head speeds for quasi-static DENT tests [20–22]. The load–displacement diagrams in DENT tests are presented in Fig. 7. Comparison of the stress–strain curves for notched and un-notched specimens is performed in Fig. 8, which confirms good agreement between the observations in DENT tests and tensile cyclic tests. 3. Treatment of observations Our aim is (i) to determine the specific essential work of fracture we by using the standard procedure, (ii) to develop a revised algorithm based on the observations reported in Section 2, and (iii) to compare results of calculations. 3.1. Standard approach We begin with the standard procedure of fitting observations in DENT tests, plot the specific work of fracture w versus ligament width H (Fig. 9), approximate the experimental data by Eq. (1), and obtain we ¼ 32:7 kJ=m2 . This quantity exceeds (about by twice) the specific essential works of fracture for natural and synthetic rubbers (their values belong to the interval between 10 and 20 kJ=m2 ). The latter may be explained by reinforcement of TPE with a relatively large amount of carbon black. A substantial increase in we with concentration of filler was mentioned in [40,41]. Eq. (1) is grounded on the Griffith concept of fracture, according to which (A) deformation of a sample outside a relatively small domain surrounding the ligament is merely elastic, which means that the main part of the sample does not provide contribution into the work of fracture, (B) the work of fracture equals the energy released when macro-crack propagates through the ligament and the energy dissipated in a small plastic zone surrounding it (due to formation and growth of fibrils, as well as initiation, growth, and coalescence of micro-voids and micro-cracks), (C) the energy of breakage of the ligament is proportional to its cross-sectional area, while the energy dissipated in the plastic zone is proportional to its volume. Our experimental data demonstrate that assumption A of the conventional theory is strongly violated as the hysteresis energies of un-notched samples are comparable with the works of fracture measured on DENT specimens (Figs. 6 and 9). To assess the energy dissipated in the plastic zone, additional measurements of density were performed. For this purpose, several pieces with length of about 10 mm were cut from undeformed specimens and their density was determined. Weight of the samples was measured by means of a Santorius 1602 MPA balance, whereas their volume was measured with the help

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Fig. 9. Specific work of fracture w versus ligament length H. Symbols: treatment of experimental data in DENT tests with the cross-head speed d_ ¼ 100 mm= min ( – T = 23,  – T = 60,  – T = 100 °C). Solid lines: their approximation by Eq. (1).

of pycnometer Accupyc 1330 (Micromeritics). The average (over six samples) density of the undeformed polymer composite is qundef ¼ 1:0715 g=cm3 . Afterwards, the same procedure was applied to pieces cut from the samples after their fracture in DENT tests. The average density of the polymer composite after deformation reads qdef ¼ 1:0764 g=cm3 . Although the last digit in these data may be questioned, they demonstrate a weak increase (not a decrease driven by void formation) in density that may be ascribed to stress-induced crystallization of the TPE matrix. Although formation and growth of micro-voids and micro-cracks in a domain surrounding the ligament are revealed by optical measurements, observations in dilatometric tests show that their contribution into the work of fracture may be rather small. 3.2. Revised approach To describe observations in tensile tests on DENT tests, the following hypotheses are introduced: (A) irreversible deformations occur both in the ligament and the remaining part of the sample, (B) under tension up to the breakage point, a plastic zone is formed in the vicinity of the ligament, but this domain is relatively small, and the energy dissipated in it is negligible compared with the hysteresis energy of the sample, (C) the work of fracture equals the energy necessary for breakage of the ligament and the hysteresis energy dissipated in the sample, (D) the hysteresis energy in a notched sample roughly coincides with the energy dissipated in an un-notched specimen under cyclic tensile deformation with the same cross-head speed and maximum strain as are measured far away from the ligament in a DENT test. According to these assumptions, the law of energy conservation reads

w ¼ we þ whys ;

ð6Þ

where whys is measured in independent tests on un-notched samples. To find the specific essential work of fracture we , we apply Eq. (6) to the observations depicted in Figs. 2 and 6. The results of analysis (Fig. 10) imply that we ¼ 32:9 kJ=m2 . Figs. 9 and 10 demonstrate that the energies we calculated by using these two approaches practically coincide. This may be explained by the fact that the specific hysteresis energy whys linearly depends on ligament width H (Fig. 6). This energy plays the same role in Eq. (6) as the quantity dHwp (associated with the energy stored in a plastic zone) in Eq. (1). From the standpoint of applications, the proposed approach has the following advantages: (1) Eq. (6) involves the only adjustable parameter we , which implies that it ensures higher accuracy of determining the specific essential work of fracture compared to Eq. (1), (2) to find we , it suffices to perform two tests only (on a notched and an un-notched sample). To demonstrate the latter property, we apply Eq. (6) to the observations presented in Figs. 7 and 8, calculate we , and plot the specific essential work of fracture as a function of cross-head speed d_ in Fig. 11 (with log ¼ log10 ). This figure demonstrates good agreement between the experimental data and their approximation by the equation [35]

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Fig. 10. Specific essential work of fracture we versus ligament length H. Circles: treatment of experimental data. Solid line: their approximation by Eq. (6).

_ Circles: treatment of experimental data on DENT samples with H ¼ 4:7 mm. Solid Fig. 11. Specific essential work of fracture we versus cross-head speed d. line: their approximation by Eq. (7) with we0 ¼ 2:41  102 kJ=m2 and we1 ¼ 5:19 kJ min =m3 .

_ we ¼ we0 þ we1 log d;

ð7Þ

where the coefficients we0 and we1 are found by the least-squares method. As this work focuses on the mechanical behavior of TPE composites at ambient temperature, we do not dwell on observations at elevated temperatures. It is worth, however, mentioning the experimental data reported in Fig. 9 (filled circles and asterisks), which show that the standard EWF theory becomes inapplicable at temperatures T = 60 and 100 °C for it results in negative values of we . The revised concept implies that we monotonically decreases with temperature, but remains positive. A shortcoming of the revised approach is that it requires an additional cyclic tensile test to be conducted on an unnotched specimen in order to determine the hysteresis energy whys . Our aim is to derive a constitutive model for the viscoplastic response of TPE composites that allows whys to be calculated as a function of strain rate _ and maximum strain per cycle max . 4. Constitutive model With reference to a homogenization concept [36], a carbon black-filled TPE is treated as an equivalent one-phase continuum. A non-affine incompressible network of flexible chains bridged by permanent junctions is chosen as the equivalent

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medium. The incompressibility condition for macro-deformation of the network is in accord with experimental data for Poisson’s ratio (close to 0.5) of thermoplastic elastomers [37]. The assumption that the network is permanent implies that the viscoelastic phenomena (associated with separation of active chains from their junctions and merging of dangling chains with the network [38,39]) are disregarded. 4.1. Kinematic equations for sliding of junctions In a non-affine network, junctions between chains slide with respect to their reference positions under deformation. Denote by F the deformation gradient for macro-deformation (its dependence on spatial coordinates and time t P 0 is suppressed for brevity), and by Fp the deformation gradient for sliding of junctions (the subscript index ‘‘p” is associated with plastic flow). The multiplicative decomposition formula implies that the deformation gradient for elastic deformation Fe reads

Fe ¼ F  F1 p ;

ð8Þ

where the dot stands for inner product. Differentiation of Eq. (8) with respect to time results in

dFe ¼ Le  Fe ; dt

ð9Þ

where

Le ¼ L  Fe  lp  F1 e ;

L¼

dF 1 dFp 1  F ; lp ¼  Fp : dt dt

ð10Þ

> 1 Let D ¼ 12 ðL þ L> Þ; W ¼ 12 ðL  L> Þ, De ¼ 12 ðLe þ L> e Þ; We ¼ 2 ðL e  L e Þ be rate-of-strain and vorticity tensors for macro-deformation and elastic deformation, respectively (> stands for transpose). To describe sliding of junctions, the following assumptions are introduced: (i) the rate-of-strain tensor for elastic deformation is connected with the rate-of-strain tensor for macro-deformation by the relation

De ¼ ð1  /ÞD;

ð11Þ

where / is a scalar function, and (ii) the vorticity tensor for elastic deformation vanishes

We ¼ 0:

ð12Þ

The approach based on Eqs. (11) and (12) slightly differs from the standard hypotheses in finite viscoplasticity of polymers. As there is no physically reasonable way to distinguish between elastic and plastic vorticity tensors, conventional theories (i) postulate that the vorticity tensor for plastic deformation vanishes and (ii) introduce an algebraic relation between the rateof-strain tensor for plastic deformation Dp , on the one hand, and the rate-of-strain tensor for macro-deformation D and the Cauchy stress tensor R, on the other. Derivation of the latter equation leads, however, to some complications, as the tensor Dp and the tensors D and R are defined in different bases. Within the present approach, kinematic equations are formulated for components De and We of the velocity gradient for elastic deformation. These tensors are defined in the same basis as the tensors D and R, which implies that Eq. (11) is objective. It follows from Eqs. (8)–(12) that the deformation gradient for sliding of junctions obeys the differential equation

dFp ¼ Fp  F1  ½L  ð1  /ÞD  F; dt which means that plastic flow in the equivalent network is entirely determined by history of macro-deformation. The left and right Cauchy–Green tensors for elastic deformation are given by

Be ¼ Fe  F>e ;

Ce ¼ F>e  Fe :

ð13Þ

The incompressibility conditions for macro-deformation and sliding of junctions together with Eq. (8) imply that the third principal invariant of Ce equals unity. Its first principal invariant reads J e1 ¼ Ce : I, where I is the unit tensor, and the colon stands for convolution. Differentiating this equality with respect to time and using Eqs. (11) and (13), we obtain

dJ e1 ¼ 2ð1  /ÞBe : D: dt

ð14Þ

4.2. Stress–strain relations The strain energy density of an equivalent network of flexible chains is given by

W eq ¼

1 lðJe1  3Þ; 2

ð15Þ

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where l stands for rigidity. Differentiating Eq. (15) with respect to time and using Eq. (14), we find that dW eq =dt ¼ lð1  /ÞBe : D. At isothermal deformation of an incompressible medium, the Clausius–Duhem inequality reads

Q ¼

dW eq þ R0 : D P 0; dt

ð16Þ

where Q stands for internal dissipation per unit volume and unit time, and R0 denotes the deviatoric component of the Cauchy stress tensor R. Combining Eqs. (15) and (16) and assuming internal dissipation to vanish, we arrive at the stress–strain relation

R ¼ pI þ lð1  /ÞBe ;

ð17Þ

where p stands for an unknown pressure. The neglect of Q in derivation of Eq. (17) means that internal dissipation per unit volume and unit time is small compared with an appropriate power of external forces R0 : D. This hypothesis can be easily avoided by replacing Eq. (11) with a more sophisticated relation

De ¼ ð1  /ÞD  wR0 ;

ð18Þ

where w is an arbitrary non-negative scalar function. Repeating the above transformations, we arrive at stress–strain Eq. (17) with

Q¼

w R0 : R0 P 0: 1/

Our approach (which is equivalent to the condition w ¼ 0) results in a substantial reduction in the number of adjustable parameters without deterioration of the quality of fitting observations, see Figs. 4 and 8. 4.3. Kinetics of plastic flow To describe evolution of / with time at cyclic deformation, we distinguish (i) the first loading of a virgin material, and (ii) its unloading. The growth of / at the first loading is described by the formula

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / ¼ U½1  expða J e1  3Þ;

ð19Þ

where a and U are dimensionless constants. Eq. (19) means that /ð0Þ ¼ 0, the coefficient / monotonically increases with time and reaches its ultimate value U at large deformations.  1 Evolution of / at unloading with the strain-rate intensity D ¼ 23 D : D 2 is governed by the differential equation

d/ ¼ D½að1  /Þ2 ðJ e1  3Þ þ b/; dt

ð20Þ

where a and b are dimensionless parameters. The first term in Eq. (20) characterizes growth of / induced by strain energy, whereas the last term reflects self-acceleration of sliding at retraction. Under uniaxial tension with small strains, Eqs. (19) and (20) are transformed into the kinetic equations

d/ ~ð1  /ÞðU0  /Þ; ¼a dt

d/ ~ ~ ð1  /Þ2 þ b/; ¼a dt

ð21Þ

pffiffiffi ~ ¼ bd=dt, where  stands for tensile strain, and e denotes the elastic strain. Differ~ ¼ a 3d=dt, a ~ ¼ 3a2e d=dt, b with a ential equations (21) have a structure that is typical of evolution equations in chemical kinetics. The quadratic terms in these relations describe interactions between different species (the matrix and filler particles in a reinforced TPE), whereas the term proportional to / characterizes acceleration of the sliding process induced by inter-chain interactions in the matrix. 4.4. Adjustable parameters Eqs. (17)–(20) provide a set of stress–strain relations for an arbitrary three-dimensional deformation of a polymer composite at finite strains. These equations involve five adjustable parameters. Three of them, l; a, and U, that characterize the viscoplastic response at first loading, are independent of deformation. We suppose that a and U are independent of strainrate intensity, whereas l weakly increases with D,

l ¼ l0 þ l1 log D;

ð22Þ

where lm ðm ¼ 0; 1Þ are constant coefficients. It should be noted that Eq. (22) is not a constitutive equation, but a conventional phenomenological relation that allows the influence of strain rate on elastic modulus to be accounted for in a simplified manner (i.e. by neglecting viscoelastic effects whose proper description requires replacement of Eq. (17) with a hereditary integral equation).

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The other parameters, a and b, are independent of strain rate, but are strongly affected by deformation at the instant when the strain rate changes its sign. The influence of maximum strain per cycle on these quantities is described by the relations

log a ¼ a0  a1 logðJ e1  3Þ;

log b ¼ b0  b1 logðJ e1  3Þ;

ð23Þ

where am ; bm ðm ¼ 0; 1Þ are constants, and J e1  3 is the dimensionless strain energy at the instant when unloading starts. The material constants l0 ; l1 ; a; U; a0 ; a1 ; b0 ; b1 are found by approximation of observations in uniaxial cyclic tensile tests. 4.5. Uniaxial deformation At uniaxial tension of an incompressible medium, macro-deformation is described by the formulas

x1 ¼ kðtÞX 1 ;

1

1

x2 ¼ k 2 ðtÞX 2 ;

x3 ¼ k 2 ðtÞX 3 ;

where fX m g and fxm gðm ¼ 1; 2; 3Þ are Cartesian coordinates in the reference and actual states, respectively, and kðtÞ stands for elongation ratio. Transformation of initial into intermediate state (that describes sliding of junctions in an equivalent network) is determined by 1

n1 ¼ kp ðtÞX 1 ;

n2 ¼ kp 2 ðtÞX 2 ;

1

n3 ¼ kp 2 ðtÞX 3 ;

where fnm g ðm ¼ 1; 2; 3Þ are Cartesian coordinates in the intermediate state, and kp ðtÞ is a function to be found. These relations together with Eqs. (8) and (10) imply that Eq. (12) is satisfied identically, whereas Eq. (11) is transformed into the differential equation

dkp kp dk : ¼/ dt k dt

ð24Þ

Using Eqs. (13) and (17), excluding pressure p from the boundary condition at the lateral surface of a sample, and introducing the engineering tensile stress r ¼ R=k, where R is the only nonzero component of the Cauchy stress tensor R, we arrive at the formula

r¼

l kp

" ð1  /Þ

 2 # k kp :  kp k

ð25Þ

Eqs. (24) and (25) together with Eqs. (19) and (20), where J e1 ¼ ðk=kp Þ2 þ 2kp =k, describe uniaxial deformation of an equivalent network. 5. Fitting of observations Our aim now is to find adjustable parameters in the stress–strain relations by fitting the experimental data reported in Figs. 4 and 8. Each set of observations is approximated separately. 5.1. Loading We begin with fitting the loading path of the stress–strain diagram with the cross-head speed d_ ¼ 100 mm/min and the maximum strain max ¼ 0:75 (unfilled circles in Fig. 4). We fix some intervals ½0; a  and ½0; U , where the best-fit parameters a and U are assumed to be located, and divide these intervals into J ¼ 10 sub-intervals by the points aðiÞ ¼ iDa and UðjÞ ¼ jDU with Da ¼ a =J; DU ¼ U =J, and i; j ¼ 0; 1; . . . ; J  1. For each pair faðiÞ ; UðjÞ g, the function kp is determined by Eq. (24),

dkp kp ¼/ ; dk k

ð26Þ

where / is given by Eq. (19). Integration of Eq. (26) is performed by the Runge–Kutta method with the step Dk ¼ 5:0  104 . The modulus l is found by the least-squares technique from the condition of minimum of the function

F¼

X

2

½rexp ðkm Þ  rnum ðkm Þ ;

ð27Þ

m

where summation is performed over all elongation ratios km ¼ 1 þ m at which observations are reported, rexp is the engineering tensile stress measured in the test, and rnum is given by Eq. (25). The best-fit values of a and U are found from the condition of minimum of function (27). Afterwards, the initial intervals are replaced with the new intervals ½a  Da; a þ Da; ½U  DU; U þ DU, and the calculations are repeated. The parameters a and U are listed in Table 1. The loading paths of the stress–strain curves depicted in Fig. 8 are approximated by using the above algorithm with the only adjustable parameter l (the other parameters are given in Table 1). When the elastic modulus l is found by matching observations at tension with various strain rates _ , the dependence lð_ Þ is plotted in Fig. 12. This figure demonstrates good

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Table 1 Adjustable parameters for TPE composite. Parameter

Dimension

Value

l0 l1

MPa MPa

4.00 0.35 8.50 0.56 1.42 0.91 0.99 1.12

a

U

a0 a1 b0 b1

Fig. 12. Elastic modulus l versus strain rate _ . Circles: treatment of experimental data in cyclic tensile tests with various cross-head speeds. Solid line: their approximation by Eq. (22).

agreement between the experimental data and their approximation by Eq. (22), where we set (approximately) D ¼ j_ j and calculate lm ðm ¼ 0; 1Þ by the least-squares technique (these coefficients are collected in Table 1). 5.2. Unloading To approximate the unloading paths of the stress–strain diagrams, Eqs. (19) and (26) are integrated from k ¼ 1 to k ¼ kmax , where kmax ¼ 1 þ max , with the adjustable parameters reported in Table 1. First, the observations are fitted in tests with the cross-head speed d_ ¼ 100 mm= min (Fig. 4). We fix some intervals ½0; a  and ½0; b , where the best-fit parameters a and b are assumed to be located, and divide these intervals into J ¼ 10 sub-intervals by the points aðiÞ ¼ iDa and bðjÞ ¼ jDb with Da ¼ a =J; Db ¼ b =J, and i; j ¼ 0; 1; . . . ; J  1. For each pair faðiÞ ; bðjÞ g, the functions kp and / are determined from Eqs. (20) and (24) that are presented in the form

dkp kp ¼/ ; dk k

h i d/ ¼  aðJ e1  3Þð1  /Þ2 þ b/ : dk

ð28Þ

Integration of Eq. (28) is carried out from k ¼ kmax to k = 1 by the Runge–Kutta method with the step jDkj ¼ 5:0  104 . The parameters a and b are found from the condition of minimum of function (27), where rnum is determined by Eq. (25). Afterwards, the initial intervals are replaced with the new intervals ½a  Da; a þ Da; ½b  Db; b þ Db, and the calculations are repeated. After finding a and b for each stress–strain diagram depicted in Fig. 4, these quantities are plotted versus the dimensionless strain energy J e1  3 at the instant when unloading starts (unfilled circles in Fig. 13). The experimental data are approximated by Eq. (23), where am and bm ðm ¼ 0; 1Þ are calculated by the least-squares method (these coefficients are collected in Table 1). Finally, the above algorithm is applied to match the experimental data reported in Fig. 8. For each set of observations, the best-fit parameters a and b are determined and plotted in Fig. 13 (filled circles). Fig. 13 demonstrates an acceptable agreement between the results on numerical analysis (for all stress–strain diagrams presented in Figs. 4 and 8) and their approximation by Eq. (23).

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1991

Fig. 13. Parameters a and b versus dimensionless strain energy Je1  3 at the instant when unloading starts. Symbols: treatment of experimental data in cyclic tensile tests. Unfilled circles: the cross-head speed d_ ¼ 100 mm= min. Filled circles: various cross-head speeds. Solid lines: their approximation by Eq. (23).

To verify the constitutive equations, the mechanical response of TPE specimens in uniaxial cyclic tensile tests is determined numerically. Appropriate stress–strain curves are plotted in Figs. 4 and 8 (solid lines). These figures show excellent agreement between the results of numerical simulation and the observations. 5.3. Numerical simulation To evaluate ability of the constitutive model to describe the viscoelastic behavior of the TPE composite at higher strain rates and to assess the ligament width at which Eq. (1) is fulfilled, numerical simulation is performed of the stress–strain relations for uniaxial cyclic tensile tests with the strain rate _ ¼ 1:0 s1 and various maximum strains per cycle. The results of analysis are presented in Fig. 14, where the engineering tensile stress r is plotted versus tensile strain . Using these data, the hysteresis energy W hys is calculated and plotted versus maximum strain per cycle max in Fig. 15 (circles). This figure demonstrates that the dependence W hys ðmax Þ can be correctly approximated by the linear function

W hys ¼ W hys 0 þ W hys 1 max ;

Fig. 14. Engineering stress various maximum strains.

ð29Þ

r versus tensile strain . Solid lines: results of numerical simulation for cyclic tensile tests with the strain rate _ ¼ 1:0 s1 and

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Fig. 15. Hysteresis energy W hys versus maximum tensile strain per cycle max . Circles: results of numerical simulation for tensile tests with the strain rate _ ¼ 1:0 s1 . Solid line: their approximation by Eq. (29) with W hys 0 ¼ 0:62 J and W hys 1 ¼ 3:35 J.

provided that the maximum strain max exceeds 0.2. Assuming dependence (4) of the maximum strain on ligament width to remain valid, we conclude from Figs. 5 and 15 that Eq. (1) may be used for the analysis of observations in DENT tests with the strain rate _ ¼ 1:0 s1 when H > 0:25h. 6. Concluding remarks Observations have been reported (i) in DENT tests with various strain rates and (ii) in tensile cyclic tests with various maximum strains on carbon black-filled thermoplastic elastomer Thermoplast K at ambient temperature. It is shown that the stress–strain diagrams in DENT tests (measured relatively far away from the ligament) coincide with those in tensile cyclic tests on un-notched specimens. The specific essential work of fracture has been determined by means of the conventional method and with the help of a revised EWF theory. The latter concept disregards the energy dissipated in a plastic zone in the close vicinity of the ligament, but accounts for the hysteresis energy in the main part of a DENT sample. It is demonstrated that values of we calculated by using these two approaches practically coincide. Based on the experimental data, it is shown that linear dependence (1) of the specific work of fracture w on ligament width H follows from the linear dependence of the maximum strain at fracture max on H, on the one hand, and the linear dependence of the hysteresis energy whys on max , on the other. According to the revised EWF concept, it suffices to perform two tests to find we : the standard test on a DENT specimen, and a tensile cyclic test on an un-notched specimen. To avoid additional experiments, a constitutive model has been developed in cyclic viscoplasticity of polymer composites, and its adjustable parameters have been determined by fitting the observations. The modified EWF theory has been applied to analyze the effect of cross-head speed on the specific essential work of fracture. It is shown that we weakly (logarithmically) grows with strain rate. A range of ligament widths, at which the specific work of fracture w remains proportional to ligament width H has been assessed by using numerical simulation. It is revealed that Eq. (1) is satisfied with an acceptable level of accuracy when the maximum tensile strain far away from the notch exceeds 0.2. Acknowledgement This work was supported by The Danish Energy Authority through Projects ENS-33033-0096 and ENS-63011-0068. Appendix A An anonymous reviewer of this paper has raised a question regarding compatibility of phenomenological Eq. (22) that presumes l to depend on strain rate intensity with stress–strain relation (17) developed from the Clausius–Duhem inequality (16) under the assumption that the elastic modulus remains constant. The aim of this appendix is two-fold: (i) to derive an analog of Eq. (17) from Eq. (16) assuming l to be an arbitrary function of time lðtÞ, and (ii) to show that this analog is transformed into Eq. (17) when l is a constant.

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1993

To rearrange Eq. (17), we introduce the extra stress

T ¼ ð1  /ÞS;

S ¼ lBe ;

ðA:1Þ

and write

R ¼ pI þ T:

ðA:2Þ

The symbol T is used instead of R0 because we do not presume T to be traceless. It follows from Eqs. (9) and (13) that the tensor Be obeys the differential equation

dBe ¼ Le  Be þ Be  L>e : dt

ðA:3Þ

Combination of Eqs. (A.1) and (A.3) implies that the tensor function SðtÞ is governed by the equation

Sr ¼ 0;

ðA:4Þ

where

Sr ¼

dS  Le  S  S  L>e dt

ðA:5Þ

stands for the Oldroyd derivative. Eqs. (A.1), (A.2), and (A.5) serve as stress–strain relations for a polymer network with a time-independent elastic modulus. To derive analogs of these equations for an equivalent network with a time-dependent modulus, we consider two cases separately, when the function lðtÞ increases and decreases, and discuss the general case afterwards. A.1. A network with a growing modulus First, we analyze the case when the function lðtÞ increases in an interval ½0; t0 , and its derivative with respect to time is positive, l_ ðtÞ > 0. Introducing the relative deformation gradient for elastic deformation

f e ðt; sÞ ¼ Fe ðtÞ  F1 e ðsÞ

ðA:6Þ

and using Eq. (9), we find that the function f e ðt; sÞ satisfies the equation

@f e ðt; sÞ ¼ Le ðtÞ  f e ðt; sÞ; @t

f e ðs; sÞ ¼ I:

ðA:7Þ

It follows from Eq. (A.7) that the relative left Cauchy–Green tensor >

be ðt; sÞ ¼ f e ðt; sÞ  f e ðt; sÞ

ðA:8Þ

is governed by the equation similar to Eq. (A.3),

@be ðt; sÞ ¼ Le ðtÞ  be ðt; sÞ þ be ðt; sÞ  L>e ðtÞ; @t

be ðs; sÞ ¼ I:

ðA:9Þ

According to Eqs. (11) and (A.9), the first principal invariant je1 of the tensor be obeys the differential equation analogous to Eq. (14),

@je1 ðt; sÞ ¼ 2ð1  /ðtÞÞbe ðt; sÞ : DðtÞ: @t

ðA:10Þ

The strain energy density of an equivalent network with a time-dependent modulus is given by

W eq ðtÞ ¼

1 2



lð0ÞðJe1 ðtÞ  3Þ þ

Z

t



l_ ðsÞðje1 ðt; sÞ  3Þds :

ðA:11Þ

0

Differentiating Eq. (A.11) with respect to time and using Eqs. (14) and (A.10), we obtain

  Z t dW eq l_ ðsÞbe ðt; sÞds : DðtÞ: ðtÞ ¼ ð1  /ðtÞÞ lð0ÞBe ðtÞ þ dt 0 Inserting this expression into Eq. (16) and disregarding internal dissipation, we arrive at Eq. (A.2), where T is determined by Eq. (A.1), and

SðtÞ ¼ lð0ÞBe ðtÞ þ

Z

t

l_ ðsÞbe ðt; sÞds: 0

ðA:12Þ

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Differentiation of Eq. (A.12) with respect to time and use of Eqs. (A.3) and (A.9) result in the formula

dS ðtÞ ¼ l_ ðtÞI þ Le ðtÞ  dt



lð0ÞBe ðtÞ þ

Z

t





l_ ðsÞbe ðt; sÞds þ lð0ÞBe ðtÞ þ

0

Z 0

t



l_ ðsÞbe ðt; sÞds  L>e ðtÞ:

This equality together with Eq. (A.12) implies that the function SðtÞ obeys the equation

Sr ¼ l_ ðtÞI:

ðA:13Þ

A.2. A network with a decreasing modulus We now focus on the case when the function lðtÞ decays in an interval ½0; t0 , and its derivative with respect to time is negative, l_ ðtÞ < 0. The only difference compared with the case l_ ðtÞ > 0 consists in the expression for the strain energy density of the equivalent network

W eq ðtÞ ¼

1 2



lðt0 ÞðJe1 ðtÞ  3Þ 

Z

t0



l_ ðsÞðje1 ðt; sÞ  3Þds :

ðA:14Þ

t

Eq. (A.11) cannot be applied as it may lead to negative values of the stored energy due to the sign of l_ ðtÞ. Repeating the above transformations, we arrive at the stress–strain relations (A.1), (A.2), and (A.13). A.3. A network with a non-monotonically changing modulus If the function lðtÞ is not monotonic, it suffices to split the entire interval of time into subintervals where it either increases or decreases and use expressions (A.11) and (A.14) in appropriate subintervals (adding some constants to W eq if necessary to ensure continuity of the strain energy density). Although different expressions are used for the mechanical energy in different intervals, the viscoplastic response of an equivalent network is governed by the same stress–strain relations (A.1), (A.2), and (A.13). When macro-deformation occurs with a constant strain-rate intensity (as in the experiments described in Section 2), Eq. (22) implies that l is independent of time, and Eq. (A.13) coincides with Eq. (A.4) employed in approximation of observations in Section 5. This confirms applicability of Eqs. (17)–(20) together with Eq. (22) for the analysis of experimental data. It is worth noting that we treat the strain energy density in the sense in which thermodynamic potentials are conventionally considered in statistical physics, i.e. as a non-negative functional on trajectories of a dynamical system. This does not cause difficulties when Eq. (A.11) is applied, but may seem a bit ‘‘exotic” when Eq. (A.14) is employed with W eq depending on final instant t0 . In the latter case, however, Eq. (15) with a time-dependent modulus does not violate the second law of thermodynamics: for a monotonically decreasing function lðtÞ, substitution of expression (15) into Eq. (16) results in the stress–strain relation (17) with dissipation rate Q ¼  12 l_ ðJ e1  3Þ P 0. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

Broberg KB. Critical review of some theories in fracture mechanics. Int J Fract Mech 1968;4:11–9. Cotterell B, Reddel JK. The essential work of plane stress ductile fracture. Int J Fract 1977;13:267–77. Mai Y-W, Cotterell B. On the essential work of ductile fracture in polymers. Int J Fract 1986;32:105–25. Mai Y-W, Powell P. Essential work of fracture and J-integral measurements for ductile polymers. J Polym Sci B: Polym Phys 1991;29:785–93. Cotterell B, Atkins AG. A review of the J and I integrals and their implications for crack growth resistance and toughness in ductile fracture. Int J Fract 1996;81:357–72. Grellmann W, Lach R, Seidler S. Experimental determination of geometry-independent fracture mechanics values J, CTOD and K for polymers. Int J Fract 2002;118:9–14. Hashemi S. Determination of the fracture toughness of polybutylene terephthalate (PBT): effect of specimen size and geometry. Polym Engng Sci 2000;40:798–808. Poon WKY, Ching ECY, Cheng CY, Li RKY. Measurement of plane stress essential work of fracture (EWF) for polymer films: effects of gripping and notching methodology. Polym Testing 2001;20:395–401. Light ME, Lesser AJ. Effect of test conditions on the essential work of fracture in polyethylene terephthalate film. J Mater Sci 2005;40:2861–6. Kwon HJ, Jar P-YB. Toughness of high-density polyethylene in plane-strain fracture. Polym Engng Sci 2006;46:1428–32. Kwon HJ, Jar P-YB. Application of essential work of fracture concept to toughness characterization of high-density polyethylene. Polym Engng Sci 2007;47:1327–37. Duan K, Hu X. Analysis of non-linear relationship between specific work of fracture and ligament of polymeric materials using modified EWF model. Key Engng Mater 2006;312:65–70. Duan K, Hu X, Stachowiak G. Modified essential work of fracture model for polymer fracture. Compos Sci Technol 2006;66:3172–8. Hashemi S. Temperature and deformation rate dependence of the work of fracture in polycarbonate (PC) film. J Mater Sci 2000;35:5851–6. Fayolle B, Tcharkhtchi A, Verdu J. Temperature and molecular weight dependence of fracture behaviour of polypropylene films. Polym Testing 2004;23:939–47. Chen H, Wu J. Specific essential work of fracture of polyurethane thin films with different molecular structures. J Polym Sci B: Polym Phys 2007;45:1418–24. Arkhireyeva A, Hashemi S. Effect of temperature on work of fracture parameters in poly(ether–ether ketone) (PEEK) film. Engng Fract Mech 2004;71:789–804. Kuno T, Yamagishi Y, Kawamura T, Nitta K. Deformation mechanism under essential work of fracture process in polycyclo-olefin materials. Express Polym Lett 2008;2:404–12. Hashemi S. Fracture toughness evaluation of ductile polymeric films. J Mater Sci 1997;32:1563–73.

A.D. Drozdov et al. / Engineering Fracture Mechanics 76 (2009) 1977–1995

1995

[20] Maspoch ML, Ferrer D, Gordillo A, Santana OO, Martinez AB. Effect of the specimen dimensions and the test speed on the fracture toughness of iPP by the essential work of fracture (EWF) method. J Appl Polym Sci 1999;73:177–87. [21] Ching ECY, Poon WKY, Li RKY, Mai Y-W. Effect of strain rate on the fracture toughness of some ductile polymers using the essential work of fracture (EWF) approach. Polym Engng Sci 2000;40:2558–68. [22] Karger-Kocsis J, Barany T. Plane-stress fracture behavior of syndiotactic polypropylenes of various crystallinity as assessed by the essential work of fracture method. Polym Engng Sci 2002;42:1410–9. [23] Grein C, Plummer CJG, Germain Y, Kausch H-H, Beguelin P. Essential work of fracture of polypropylene and polypropylene blends over a wide range of test speeds. Polym Engng Sci 2003;43:223–33. [24] Chen H, Karger-Kocsis J, Wu J. Effects of molecular structure on the essential work of fracture of amorphous copolyesters at various deformation rates. Polymer 2004;45:6375–82. [25] Pegoretti A, Ricco T. On the essential work of fracture of neat and rubber toughened polyamide-66. Engng Fract Mech 2006;73:2486–502. [26] Karger-Kocsis J, Czigany T, Moskala EJ. Thickness dependence of work of fracture parameters of an amorphous copolyester. Polymer 1997;38:4587–93. [27] Ching ECY, Li RKY, Mai Y-W. Effects of gauge length and strain rate on fracture toughness of polyethylene terephthalate glycol (PETG) film using the essential work of fracture analysis. Polym Engng Sci 2000;40:310–9. [28] Maspoch ML, Gamez-Perez J, Karger-Kocsis J. Effects of thickness, deformation rate and energy partitioning on the work of fracture parameters of uPVC films. Polym Bull 2003;50:279–86. [29] Zhao H, Li RKY. Fracture behaviour of poly(ether ether ketone) films with different thicknesses. Mech Mater 2006;38:100–10. [30] Wong S-C, Baji A, Gent AN. Effect of specimen thickness on fracture toughness and adhesive properties of hydroxyapatite-filled polycaprolactone. Composites A 2008;39:579–87. [31] Gamez-Perez J, Santana O, Martinez AB, Maspoch ML. Use of extensometers on essential work of fracture (EWF) tests. Polym Testing 2008;27:491–7. [32] Horgan CO, Knowles JK. Recent developments concerning Saint–Venant’s principle. Adv Appl Mech 1983;23:179–269. [33] Horgan CO. Recent developments concerning Saint–Venant’s principle: an update. Appl Mech Rev 1989;42:295–303. [34] Horgan CO. Recent developments concerning Saint–Venant’s principle: a second update. Appl Mech Rev 1996;49:S101–11. [35] Pegoretti A, Marchi A, Ricco T. Determination of the fracture toughness of thermoformed polypropylene cups by the essential work method. Polym Engng Sci 1997;37:1045–52. [36] Drozdov AD, Christiansen J. Cyclic thermo-viscoplasticity of carbon black-reinforced thermoplastic elastomers. Compos Sci Technol 2008;68:3114–22. [37] Kojima T, Kikuchi Y, Inoue T. Morphology-properties relationship of a ternary polymer alloy. Polym Engng Sci 2004;32:1863–9. [38] Green MS, Tobolsky AV. A new approach to the theory of relaxing polymeric media. J Chem Phys 1946;14:80–92. [39] Tanaka F, Edwards SF. Viscoelastic properties of physically cross-linked networks. Transient network theory. Macromolecules 1992;25:1516–23. [40] Huang J-C, Muangchareon P, Grossman SJ. Effects of carbon black on mechanical properties of polycarbonate–polypropylene blends. J Polym Engng 2000;20:111–20. [41] Kobayashi H, Shioya M, Tanaka T, Irisawa T, Sakurai S, Yamamoto K. A comparative study of fracture behavior between carbon black/poly(ethylene terephthalate) and multiwalled carbon nanotube/poly(ethylene terephthalate) composite films. J Appl Polym Sci 2007;106:152–60.