# Plastic deformation of low-density polyethylene reinforced with biodegradable polylactide, part 2: creep characterization and modeling.

INTRODUCTIONIn the first part of this work (1) we investigated the microstructure of several blends obtained by combining low-density polyethylene (LDPE) with biodegradable polylactide (PLA). We analyzed their heterogeneous micro-structure and compared their plastic response with that of neat LDPE and PLA under stretching at constant true strain rate by means of a video-controlled tensile testing system developed in this laboratory (2). In particular, we found that the formation of dispersed PLA nodules induces a reinforcement of the mechanical properties of the LDPE in the viscoelastic stage, while keeping reasonable ductility and toughness.

However, stretching at constant strain rate is not the only deformation mode that is found in common practice. Additional creep tests are sometimes envisaged when the interest is focused onto the deformation and fracture of materials under permanent loading. In industrial tensile-creep tests used for engineering purposes, the dumb-bell-shaped specimens are often loaded by means of a dead weight (3), (4). As such, the samples are subjected to a constant nominal stress, or Kirchhoff stress, [[sigma].sub.N] = F/[S.sub.0], where F is the applied force and [S.sub.0] the initial cross-section of the sample. The curves obtained from these tests are expressed through the evolution with time of the nominal strain, Lagrangian strain, [[epsilon].sub.N] = (L - [L.sub.0])/[L.sub.0], where [L.sub.0] and L are the initial and current values of the gauge length of the sample, obtained with an extensometer.

Obviously, nominal stress and strain represent the intrinsic behavior of the material only if the deformation is homogeneous all over the specimen. This is the reason why fundamental research has been often limited to very small strains. In previous studies on neat polyethylene (5), (6), authors have shown that, for [[epsilon].sub.N] < 0.05, the creep strain was evenly distributed, controlled by viscoelastic processes and largely reversible. Conversely, for tests performed under higher stresses with neat polyethylene and polypropylene (4), (7), it was observed that the deformation becomes more and more instable (necking) and irreversible. Eventually the strain versus time curve displays a dramatic upturn that leads to specimen rupture at a critical time. The value of this "creep lifetime" is somewhat scattered for a given material, since it depends on the distribution of defects within the sample. Nevertheless it constitutes a useful indication for characterizing the resistance to permanent loading.

In principle, for engineering applications, a whole set of creep tests should be programmed for each polymeric grade over a wide range of loads and temperatures. Since the duration of these tests is often long (typically those run under low stresses), a creep campaign uses to be very costly. Consequently, it is useful to dispose of a methodological tool capable of predicting the creep kinetics on the basis of constitutive parameters determined from tests performed during much shorter times.

The aim of this second paper is double: (i) to extend the field of application of the video-controlled tensile test to the determination of plastic behavior under creep at constant true stress, and (ii) to apply this novel creep method to the PLA/LDPE blends in order to compare quantitatively their response under constant true stress versus under constant force.

MATERIALS AND METHODS

The origin and processing conditions of the materials investigated in this work were detailed in the first part of this double article (1). Consequently, we will simply recall here their main characteristics. The LDPE (Novex 21 H 460 by Solvay) is a common industrial grade with a weight average molar mass of 117,203 g/mol, a Melt Flow Index (MFI) of 1.5 g/10 min and a crystallinity index of 42.5 wt%. The PLA (NatureWorks 3001D by Cargill-Dow) contains 92% L-lactide and 8% mesolactide; its weight average molecular mass is equal to 152,000 g/mol, its MFI is about 20 g/10 min and its crystallinity index is less than 14 wt%. Two LDPE/PLA blends were prepared in which the PLA fraction was adjusted to 10 wt% and 20 wt%, respectively. It should be noticed that in order to ensure optimal interfacial compatibility, the LDPE was partly grafted with maleic anhydride (10 wt% of the PLA content). After careful desiccation for 12 hours under vacuum at 50[degrees]C, the constituents of the blends were intimately mixed together using a corotating twin-screw extruder (Clextral BC 21) operating between 165[degrees]C and 180[degrees]C at 100 rpm The specimens for the mechanical tests were cut out of 4 mm thick plates injected with a Sandretto hydraulic press (Euromap 310/95) in a temperature range from 170[degrees]C to 195[degrees]C, with an injection pressure of 70 bars.

The specimens for the creep tests are the same as those utilized for the tensile tests at constant true strain rate. In both cases, the true strains (Hencky strains) along the axial direction ([[epsilon].sub.33]) and the transverse directions ([[epsilon].sub.11] and [[epsilon].sub.22]) was assessed in a prescribed region of the sample by means of the video-controlled testing system invented by two of us (8) and extensively analyzed in the first article (1). The system utilized in this work (VideoCreep[C]), is an optional feature of the general VideoTraction[C] system developed by Apollor S.A. (Moncel-les-Luneville, France). Since this system avoids the utilization of mechanical extensometers, it makes possible to perform the creep tests at elevated temperatures in an environmental chamber equipped with optical windows. For the present application, we applied two different creep procedures.

In the first creep procedure, the axial true stress (Cauchy stress: [[sigma].sub.33] = F/S) is kept constant throughout the experiment in the same region as for the true strains. The actual cross-section is assesses in real time by the relation S = [S.sub.0] exp([[epsilon].sub.11] + [[epsilon].sub.22]), where [S.sub.0] is the initial cross-section measured carefully prior the test in the region of interest, and ([[epsilon].sub.11], [[epsilon].sub.22]) are the transversal strains. This protocol is interesting since it has been stated by previous authors (for example, see Ref. 9) that the nonhomogeneous decrease of the cross-section during the creep tests provokes a progressive increase of the true stress that cannot be deduced simply from the nominal stain. Here, the force applied by the actuator, F, is automatically reduced by the system as S decreases. The results of these tests are displayed in terms of the evolution of the axial true strain, [[epsilon].sub.33], versus the elapsed time. Although we published our system several years ago (10), it is only very recently that a concurrent team (11) has developed an alternative system in order to control creep tests at constant local true stress, by means of a laser beam extensometer. However, that system is somewhat hybrid since the creep deformation is merely determined as the nominal strain and not through the local true strain.

In another procedure, we simply reproduce the normalized tests (3) for comparison. In that scope, the nominal stress ([[sigma].sub.N] = F/[S.sub.0]) is kept constant by simply fixing the applied forced by means of the built-in load regulator of the tensile machine. The results of these tests are displayed in terms of the evolution of the nominal strain [[epsilon].sub.N] = (L - [L.sub.0])/[L.sub.0] versus time. The tests are stopped at the time of rupture, [t.sub.R].

CREEP BEHAVIOR AT CONSTANT TRUE STRESS

The [[epsilon].sub.33](t) curves displayed in Fig. 1 were obtained with the three materials under investigation from creep tests performed at 23[degrees]C and 50[degrees]C, while the true stress was regulated at 7.2 MPa and 4.0 MPa, respectively. It is interesting to note that, at both temperatures and for the three materials, the creep rate shows a gradual decrease as the creep deformation proceeds. This behavior contrasts with the strain-rate acceleration (or "tertiary creep") observed in polymer samples subjected to tensile creep under constant force (4), (7). The absence of this ultimate stage in the present experiments is simply due to the fact that the true stress is kept constant and not the nominal stress. This decrease of strain rate during creep is clearly ascribed to the marked strain hardening in the plastic stages of LDPE and LDPE/PLA blends, revealed in the first part from tests at constant true strain rate (1).

[FIGURE 1 OMITTED]

For T = 23[degrees]C and [[sigma].sub.33] = 7.2 MPa (Fig. 1a), the creep curve for neat LDPE levels off at [[epsilon].sub.33] = 0.19, that is 27% higher than for LDPE/10% PLA ([[epsilon].sub.33] = 0.15) and 58% than for LDPE/20% PLA ([[epsilon].sub.33] = 0.12). These results indicate clearly that the addition of PLA in the LDPE decreases significantly its creep compliance. At T = 50[degrees]C and [[sigma].sub.33] = 4 MPa (Fig. lb), the same tendency is observed, but the strain plateau for neat LDPE ([[epsilon].sub.33] = 0.14) is only 5 % higher than that of LDPE/10% PLA ([[epsilon].sub.33] = 0.133) and 17% than that of LDPE/20% PLA ([[epsilon].sub.33] = 0.12). As such, the effect of blending on the creep compliance decreases when temperature is increased.

Let us now analyze the creep behavior at constant true stress in terms of a phenomenological equation, [[epsilon].sub.33] ([[epsilon].sub.33], [[sigma].sub.33]), that relates the three state variables of the material at each temperature. The main interest of this equation, if it is validated, is to correlate the long-term creep behavior with the equation [[sigma].sub.33] ([[epsilon].sub.33], [[epsilon].sub.33]) obtained from much shorter tests at constant true strain rate by means of the VideoTraction[C] system. Such a prediction is even more interesting if the constitutive relation can be expressed in a simple analytic form. Here, we model the behavior of the three materials on the bases of the pioneer work by G' Sell and Jonas (12) using a multiplicative equation: [[sigma].sub.33] =[Florin] ([[epsilon].sub.33]) X g([[epsilon].sub.33]). Such a relationship was verified with a number of polymers (13), (14). In this constitutive relation, the first term is written as [Florin] ([[epsilon].sub.33]) = K * Y([[epsilon].sub.33]) * H([[epsilon].sub.33]) where K is a scaling factor sometimes referred as the "consistency factor" (15), Y([[epsilon].sub.33]) represent the viscoelastic response up to the yield point, and H([[epsilon].sub.33]) is the strain-hardening function that describes the progressive consolidation induced by plastic deformation. In the case of semicrystalline polymers with a rubber-like amorphous phase, a Maxwell-type viscoelastic expression Y([[epsilon].sub.33]) = [1 - exp(-w * [[epsilon].sub.33])] is utilized and the strain-hardening is better described by the function H([[epsilon].sub.33]) = exp(h * [[epsilon].sub.33.sup.n]) (13). Concerning the second term, g([[epsilon].sub.33]), it is classically expressed by a power law, g([[epsilon].sub.33]) = [[epsilon].sub.33.sup.m], where m is referred as the "strain-rate sensitivity coefficient."

As such, the constitutive equation that supposedly represents the mechanical behavior of the polymers under investigation is preferentially written in two alternative forms according to its utilization for modeling the behavior at constant true strain rate or at constant stress, respectively:

[[sigma].sub.33] = K * [1 - exp(-w * [[epsilon].sub.33])] * exp(h * [[epsilon].sub.33.sup.n]) * [[epsilon].sub.33.sup.m], (1)

[[epsilon].sub.33] = [[[[[sigma].sub.33]/[K * [1 - exp(-w * [[epsilon].sub.33])] * exp(h * [[epsilon].sub.33.sup.n])]]].sup.[1/m]] (2)

In the matter that follows, we will check the validity of this law and determine the most suitable set of parameters (K, w, h, n, and m) for the three materials.

The fitting procedure does not need any sophisticated software. Like in a previous work (13), we utilized linear regression adjustment with a standard datasheet (Excel[C] or other). In a first step, the strain-rate sensitivity coefficient, m, is obtained for fixed values of [[epsilon].sub.33] from log ([[sigma].sub.33]) versus log ([[epsilon].sub.33]) plots obtained either from tensile tests at constant true strain rate (1), or from creep tests at constant true stress (this work). The second step focuses on the viscoelastic relation, ([[sigma].sub.33]/[[epsilon].sub.33.sup.m] = K * [1 - exp(-w. [[epsilon].sub.33])], that is followed at small strains when exp (h * [[epsilon].sup.n]) is close to 1. As such, K and w are essentially obtained from the preyield part of the [[sigma].sub.33] ([[epsilon].sub.33]) curves obtained at fixed values of [[epsilon].sub.33] (1). Finally the strain hardening behavior, described by the function exp (h * [[epsilon].sup.n]), is analyzed form the large-strain part of the [[sigma].sub.33] ([[epsilon].sub.33]) curves (1). The best constitutive parameters thus obtained, K, w, h, and n, are displayed in Table 1.

TABLE 1. Experimental parameters of the constitutive equation. Formulations T ([degrees]C) K (MPa) w h n m Neat LDPE 23 10.0 31 1.7 1.10 0.052 LDPE/10% PLA 23 12.3 37 1.9 1.25 0.046 LDPE/20% PLA 23 18.0 68 2.1 1.40 0.062 Neat LDPE 50 5.4 21 1.7 1.10 0.020 LDPE/10% PLA 50 7.5 23 1.9 1.25 0.042 LDPE/20% PLA 50 8.9 58 2.1 1.40 0.060

In order to evaluate the general quality of this modeling, it is interesting to analyze firstly the graph in Fig. 2 that collects data obtained from all experimental data from tensile tests at constant [[epsilon].sub.33] and from creep tests at constant [[sigma].sub.33] For each temperature (23[degrees]C and 50[degrees]C), the plots show the evolution of Ln([[sigma].sub.33]) - (h * [[epsilon].sub.33.sup.n]) versus Ln ([[epsilon].sub.33]) for the three materials under investigation here (neat LDPE, LDPE/10% PLA, and LDPE/20% PLA). It is evident As one remarks, the data coming from both testing procedures are satisfactorily aligned for each material. This result proves that, in the limit of experimental errors, the strain-rate sensitivity coefficients, m, reported in Table 1 are valid for tensile and creep tests.

[FIGURE 2 OMITTED]

As for the modeling of strain hardening through the exponential variation (exp(h * [[epsilon].sup.n])), it is better evaluated from the tensile tests at constant [[epsilon].sub.33]. Figure 3, for the example of neat LDPE, displays together the experimental and theoretical [[sigma].sub.33] ([[epsilon].sub.33]) curves at 23[degrees]C and 50[degrees]C under a fixed strain-rate, [[epsilon].sub.33] = [10.sup.-3] [s.sup.-1]. It is evident that the constitutive equation is correct for true strains up to about 0.5. At higher strains, a progressive discrepancy appears, since the experimental curves show significant softening while the theoretical hardening goes on increasing indefinitely. This discrepancy, which is also observed for the LDPE/PLA blends, is due to the damage processes that develop at large strains, particularly the diffuse cavitation that was assessed in the first part of this article (1). We consider that the theoretical curves in Fig. 3 depict the intrinsic stress versus strain curves that would be undergone by the materials in the absence of deformation damage. Whatever the detailed nature of the damage mechanisms that occur at very large strain in tensile tests, they have a minor influence on the creep tests that are limited to strains lower than [[epsilon].sub.33] = 0.6.

[FIGURE 3 OMITTED]

This phenomenological modeling provides interesting information on the influence of PLA addition on the LDPE properties. The first point is the large increase of the viscoelastic coefficient, w, which indicates that the incorporation of PLA stiffens considerably the LDPE in the viscoelastic range and consequently slows down the transient creep rate. Secondly, the dramatic increase of the consistency factor, K (that triples more or less for an addition of 20 wt% PLA) characterizes the higher resistance of LDPE/PLA blends to plastic flow and to steady-state creep. Thirdly, the increase of the strain-hardening phenomenon (increase of h and n) corresponds to the more difficult development of shear bands between the dispersed PLA particles in the LDPE matrix. Lastly, according to the literature (13), the increase of the strain-rate sensitivity coefficient, m, induces a certain stabilization of plastic deformation when PLA is added to LDPE.

The results obtained in this work show that elevating the temperature from 23[degrees]C to 50[degrees]C increases the creep compliance of the blends considerably (Fig. 1). This is due to the fact that the neat PLA, gains considerable ductility on heating. As we showed in the first article (1), this glassy material breaks in the elastic range when stretched at room temperature and, by contrast, it reaches a strain of [[epsilon].sub.33] = 0.8 at 50[degrees]C.

The main conclusion of this section is that the constitutive relation determined from the tensile tests performed at constant true strain rate remains valid for other loading histories. We will now apply it to the prediction of standard creep tests at constant force.

CREEP BEHAVIOR AT CONSTANT FORCE

In a creep test at constant force, F (that is at constant nominal stress, [[sigma].sub.N] = F/[S.sub.0]) the true stress, [[sigma].sub.33] = F/S, increases gradually as the current cross-section decreases (9), (10). If we neglect the density variations of the material (that is verified at moderate strains) we simply write: [[sigma].sub.33]([[epsilon].sub.33]) = [[sigma].sub.N] X exp([[epsilon].sub.33]). For a standard creep test at constant force, the inverse G'Sell-Jonas law thus gives:

[[epsilon].sub.33]([[epsilon].sub.33]) = [[[[[[sigma].sub.N]exp([[epsilon].sub.33])]/[K * (1 - exp(-w * [[epsilon].sub.33]) * exp(h * [[epsilon].sub.33.sup.n])]]].sup.[1/m]] (3)

The simple integration of this differential equation gives the evolution of true strain versus time. Figure 4 shows the [[epsilon].sub.33](t) obtained from this model at 23[degrees]C and 50[degrees]C, for [[sigma].sub.N] = 7.2 MPa and 4 MPa, respectively. By contrast to the curves obtained at constant true stress (Fig. 1), we see now that the strain rate increases rapidly at large strain under constant force. This result shows that the so-called "tertiary creep" phenomenon merely reflects the increase of the true stress as creep deformation proceeds.

[FIGURE 4 OMITTED]

Whatever the fundamental interest of modeling the evolution of true strain versus time, it is important to connect this approach to the standard creep tests currently utilized for plastics in industry (3). In order to dispose on an experimental data base for this study, we have tested the three materials at 23[degrees]C under investigation to three constant nominal stresses: [[sigma].sub.N] = 7.5 MPa, 8.5 MPa, and 9 MPa. The nominal creep curves obtained from these tests are displayed in Fig. 5. On the overall, they exhibit a slow creep rate at short times followed by a dramatic acceleration that terminates at specimen rupture. We examine below how our model predicts the evolution of[[epsilon].sub.N](t) in such tests. As long as the deformation is homogeneous, this variable is simply connected with the true strain by the relation: [[epsilon].sub.N](t) = exp[[[epsilon].sub.33](t)] - 1. In addition to the experimental curves, the graphs in Fig. 5 show the theoretical curves predicted with our model. It can be noticed that the creep kinetics is predicted with a precision better than 20%, which is quite encouraging considering the simple simulation scheme adopted.

[FIGURE 5 OMITTED]

Residual discrepancies are observed in the viscoelastic stage and at large strains. The former case may be associated to two causes: (i) the insufficiently precise loading procedure that brings some perturbation at small creep times, (ii) the imperfect description of the viscoelastic response in the constitutive relation, where the single parameter w of the Maxwell-type law does not reflect completely the rheological complexity of the materials. As for the ultimate deformation stage, the simulations do not explicitly take into account the cavitation phenomenon revealed with our testing method. Consequently, in most cases, the model lags a little bit behind the actual behavior at large strains. Further efforts in these different domains will be paid for getting even better predictions.

For practical use, it would be interesting to implement this model at arbitrarily chosen temperatures. In that scope, it would be necessary to determine the dependence of the constitutive parameters with temperature: K(T), w(T), h(T), n(T), and m(T). Since our tests were performed at two temperatures only, it is not reasonable to envisage a detailed functional analysis, except perhaps for the viscoelastic stage. From the indications given by the literature (15), (16), K(T) can be fitted the Arrhenius law and w(T) with the Williams-Landel-Ferry (WLF) equation. Whatever, for practical application of our model in a temperature range centered on the 23[degrees]C/50[degrees]C interval, a simple approach could consist to fit the variations with T by means of a linear interpolation.

CONCLUSIONS

In this two-part work devoted to LDPE/PLA blends, we have shown how the mechanical properties of a commodity plastic (LDPE) are improved by the incorporation of defined amounts of a highly glassy polymer (PLA). The novel testing methods (VideoTraction[C] and Video-Creep[C]) utilized here let us determine the constitutive relation of the investigated materials under three different loading conditions: (i) stretching at constant true strain rate, (ii) creep at constant true stress, and (iii) creep at constant nominal stress. Covering a wide range of strain rates, these tests are relevant to analyze various situations met in real life (accidental deformation, permanent loading in service, etc.) Also the five-parameter G'Sell-Jonas law revealed itself as a correct model to represent the experimental curves for all loading conditions. Subsequently, it could be implemented as an operational approximation in the finite-element codes currently utilized in industry.

The cavitation that was identified in the LDPE/PLA blends is an important phenomenon that was largely ignored in the past. It is essentially attributed to the partial decohesion of PLA particles from the LDPE matrix. Studying the causes and the consequences of this form of deformation damage will necessitate a special investigation. In particular, we are interested to explore more precisely: (i) how the grafting of PLA particles to LDPE could allow better control of cavitation kinetics, (ii) what are the advantages and drawbacks of cavitation in the fracture mechanics of such materials, and (iii) which modifications should be brought to the constitutive modeling to incorporate the cavitation process in view of improving the precision of creep predictions.

In addition to the above points, we will extend our study to other products based on LDPE and biodegradable materials. In particular, we are interested to valorize different types of vegetal fibers available in Algeria in the reinforcement of plastics. Beyond its practical interest, such research will address various scientific problems that did not appear with synthetic PLA, including: (i) selection of natural fibers of reproducible quality, (ii) optimization of fiber-matrix interface by chemical pretreatments, and (iii) control of fiber orientation in the forming process of the composite material.

REFERENCES

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Correspondence to: G'Sell; e-mail: gsell@mines.inpl-nancy.fr

Published online in Wiley Online Library (wileyonlinelibrary.com).

[C] 2010 Society of Plastics Engineers

F. Rezgui, (1) C. G'Sell, (2) A. Dahoun, (2) J.M. Hiver, (2) T. Sadoun (1)

(1) Departement de Genie des Procedes, Laboratoire des Materiaux Organiques, Universite de Bejaia, Faculte de la Technologie, Route de Targa Ouzemour, 06000 Bejaia, Algerie

(2) Departement SI2M, Institut Jean Lamour (UMR 7198 CNRS/Nancy-Universite/UPV-Metz), Ecole des Mines de Nancy, Parc de Saurupt, 54042 Nancy, France

DOI 10.1002/pen.21796

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Author: | Rezgui, F.; G'Sell, C.; Dahoun, A.; Hiver, J.M.; Sadoun, T. |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Geographic Code: | 6ALGE |

Date: | Jan 1, 2011 |

Words: | 4252 |

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