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Uniaxial tensile properties underlying plane-stress rapid fracture resistance in polyethylene.


Polyethylene (PE) grades designed for fuel-gas and water distribution pipe systems are rated primarily on their long-term strength, measured by hydrostatic pressure testing. Thus, ISO 12162 defines a "PE 100" resin as one which, in the form of extruded pipe, can withstand 10 MPa hoop stress for 50 years. However, a PE 100 resin must also resist failure by brittle rapid crack propagation (RCP) at 0[degrees]C and at a considerably higher equivalent pressure. Long-term failure and RCP fracture modes are so different that designing PE 100 resins with adequate resistance to both poses a serious technical challenge (1), while the competition to do so is intense.

The relatively low strength of PE necessitates a thick pipe wall, which in turn favors plane strain fracture. Unfortunately PE--like many thermoplastics--also has relatively little resistance to plane strain fracture at crack speeds, exceeding 100m/s and temperatures below 0[degrees]C. Fortunately, the plane-stress region adjacent to each free surface continues to resist separation even at high speeds and quite low temperatures. The ductility and volumetric nature of deformation here is indicated by deep stress whitening and a pronounced shear lip (see Fig. 1). Provided that, at 0[degrees]C, the plane-stress plastic zone is wide enough (a millimeter or more) and sufficiently ductile at high rates, the pipe wall thickness as a whole will be able to arrest RCP. If plane stress is suppressed, by notching (2) or otherwise, RCP resistance may be seriously reduced. The aim of the work reported here is to better understand the origin of high-rate plane-stress fracture resistance in basic tensile properties which can already, to some extent, be "designed into" a polymer.


Plane strain fracture in plastics is well characterized using concepts and methods of fracture mechanics (3), but no test method for plane-stress conditions has yet gained general acceptance. Two current methods use the familiar impact-bend configuration (see Fig. 2). The "Thin Charpy" test devised by Harry and Marshall (4) has been developed for this purpose by Brown (5) and adopted by ASTM as a standard indexing method (F 2231). The specific fracture energy from a plane strain Charpy-like test can be converted to a fracture mechanics property [G.sub.IC] as described in ISO 17281, but there is no theoretical basis for doing this from the Thin Charpy test. The Reversed Charpy test of Ritchie et al. (6) aims to reproduce the geometry, constraint, and rate under which the plane-stress zone develops. The Reversed Charpy test has not yet been proven as an indexing method but, at the cost of requiring load and displacement instrumentation, it does claim to measure a transferable material property rather than an index. This property is the plastic work dissipation, [w.sub.p], having the dimensions of energy per unit volume in recognition of the volumetric nature of plane-stress deformation (7), (8). For PE, [w.sub.p] increases with temperature and so, though the surface area through which it acts remains small, it comes to dominate the overall fracture resistance. It is this quantity, and its dependence on temperature, which will be used to characterize plane-stress fracture resistance; a quantitative scheme to do so is suggested in the Discussion.


The high-speed image of Fig. 3 illustrates that the plane-stress region in PE resists separation by drawing. In general, the work per unit volume absorbed by isothermal drawing is proportional both to the "natural" draw ratio and to the drawing stress. However, since the drawing process here id rapid, it is adiabatic. Hillmansen (7) demonstrated, by increasing the crosshead speed in a series of dog-bone tensile tests, that adiabatic heating eventually destabilized drawing. A high yield stress increases drawing work, but hastens thermal softening, destabilization, and rupture. Thus, [w.sub.p] is expected to depend not only by the rate of work absorbed in isothermal drawing but also on flow stress, thermal properties, the dependence of mechanical properties on temperature, and the proportion ([beta]) of mechanical work transformed to heat (9).


The present article models this complex interaction in a rough and ready way. The objective is to identify the deformation properties which determine [w.sub.p]. Because the properties cited are relatively fundamental in the physical sense they can in principle, be related directly to structural parameters--and, thus, can be used to guide resin development.


To study the destabilization of tensile drawing by high-rate adiabatic heating, Leevers et al. (10) measured uniaxial true stress vs. true strain properties at several strain rates and temperatures using the video extensometry method of Hiss and Strobl (11). Each stress-strain curve proved to be well represented by the Haward-Thackray spring-dashpot model--whose applicability to PE had been established by a number of previous studies (12-14). The model represents independent viscous and strain-hardening processes, acting in parallel as represented by the spring and dashpot in Fig. 4. For simplicity, elastic deformation is usually neglected, the hardening model is restricted to Gaussian rubber elasticity and the viscous element is regarded as strain-independent--i.e. Newtonian. The Gaussian elasticity is implicitly associated with a stable network of amorphous-phase entanglements or crystalline "crosslinks," and the Newtonian element with amorphous-phase molecular viscosity or with crystalline slip. It is important to recognize that this physical interpretation of the parameters is questionable, but that as an empirical representation the model works very well.


The Gaussian Haward-Thackray model (see Fig. 4) has a uniaxial true stress, [[sigma].sub.t], vs. true strain [epsilon] relationship

[[sigma].sub.t] = [[sigma].sub.y] + [G.sub.p]([[lambda].sup.2] - 1/[lambda]), (1)

where [lambda] is the extension ratio, so that [epsilon] = In [lambda]. Where true stress vs. true strain can be measured, the yield stress [[sigma].sub.y] and the strain hardening modulus [G.sub.p] can be determined from the intercept and the slope, respectively, of the [[sigma].sub.t] versus ([[lambda].sup.2] - 1/[lambda]) plot (Fig. 4b). Here [[sigma].sub.y] corresponds to the yield stress as conventionally determined for polymers, i.e., the maximum on a nominal stress versus engineering strain plot.

It can also be shown from Eq. 1 that if the material has a sufficiently high yield stress ([[sigma].sub.y] > 3[G.sub.p]) the nominal stress strain curve will subsequently pass through a minimum, and that there will be stable cold drawing at that minimum stress. This is the basis for Considere's construction method, usually applied to a maximum that corresponds to the existence of a first tangent line from the origin to the [[sigma].sub.t] versus [lambda] curve. Stable "cold" drawing corresponds to a second Considere tangent and if there is no second tangent the material will neck down to unstable failure. Hence, if the material draws at a constant stress and a natural draw ratio [[lambda].sub.d], Eq. I can be manipulated to determine [G.sub.p] as

[G.sub.p] = [[[sigma].sub.y]/[[[lambda].sub.d.sup.2] + [2/[[lambda].sub.d]]]] (2)

This is experimentally simpler than determining the true stress vs. true strain curve, and relies less on the accuracy with which the model fits measured stress-strain properties over a broader range of strain.

We now investigate the possibility that, at high strain rates, softening by adiabatic heating counteracts intrinsic strain hardening and eliminates the second tangent. In order to extrapolate to adiabatic rates, we follow Haward (13) by determining [[sigma].sub.y] and [G.sub.p] at several strain rates and temperatures, and representing the results using the Eyring flow equation. Thus, each material is initially reduced to a constitutive model of two Eyring equations, each having three parameters: an activation energy [DELTA]H, an activation volume V* and a reference strain rate [[epsilon].sub.0]:

[[[sigma].sub.p]/T] = [2/[V*.sub.[sigma]]][ - ([[DELTA][H.sub.[sigma]]]/T) + 2.303Rlog ([epsilon]/[[epsilon].sub.0[sigma]])], (3a)

[[G.sub.p]/T] = [2/[V*.sub.G]][-([[DELTA][H.sub.G]]/T) + 2.303Rlog ([epsilon]/[[epsilon]].sub.0G])]. (3b)

Note that the strain rate [epsilon] is not well determined either in a tensile drawing process or during the extension of the Reversed Charpy ligament with which it will be compared. In a tensile specimen, the effective gauge length is scaled by the total axial length of the neck shoulders. In the Reversed Charpy specimen, an effective separation rate of the same order as the impact speed (1 m/s) is imposed as the specimen halves rotate like solid blocks, while--as Fig. 3 suggests--the effective gauge length is scaled by the ligament thickness. Each of these reference dimensions is of the order of 2 mm.

Under adiabatic conditions, the increase [DELTA]H, in temperature is given as a function of strain by

[DELTA]T = [integral][beta][[[sigma].sub.y]/[[rho][C.sub.p][lambda]]]d[lambda], (4)

where [beta] is the proportion of mechanical work converted into heat (the thermomechanical conversion factor), [rho] the density of the material, and [C.sub.p] its heat capacity. Thus, a high-rate, adiabatic stress-strain curve is simulated from isothermal data by integrating, from a specified initial temperature, while continuously correcting the temperature-dependent Haward-Thackray parameters by Eyring extrapolation. The strain rate chosen should not unduly influence the result, but should be representatively "high." The argument of the previous paragraph suggests a strain rate of (1 m [s.sub.-1]/0.002 m) - 500 [s.sup.-1]. For this strain rate, the simulated stress-strain curve can then be searched for a second Considere tangent.

Unfortunately, the thermomechanical conversion factor [beta], which for PE lies in the region 0.7-0.9, is not known with any precision and is difficult to determine. Our approach is to test a series of [beta] values, both within and outside this realistic range (note that [beta] = 0 represents isothermal conditions while [beta] = 1 represents perfectly adiabatic conditions). Above some critical thermomechanical conversion factor, [[beta].sub.c] the second Considere tangent vanishes.

It is thus [[beta].sub.c], determined through an approximate analytical model from experimental data, which is used as an index to the robustness of stable drawing under fast, adiabatic conditions. Assuming that differences in [[beta].sub.c] between polyethylenes are secondary, it can be concluded that a low [[beta].sub.c] will indicate unreliable and a high [[beta].sub.c] extremely robust drawing at high rates. Indeed, if the model were accurate, a material with [[beta].sub.c] > 1 would exhibit stable drawing at any high strain rate.


Krishnaswamy et al. (15) compared the RCP performance of seven PE resins. Because these PEs represented a wide range of structures (not all were pipe grades, and their suitability for pipe ranged between excellent and very poor) the same seven materials were used for the present study. Plaques of 3 mm and 12 mm thickness were molded from each material according to ISO 292. Plaques were then remolded in a purpose-made press which imposed at each of their surfaces a controlled cooling rate of between 0.5 and 10[degrees]C/min. The 3-mm material was used to prepare ISO 527-2, type 5B dumb-bell tensile specimens. These specimens were tested at three constant temperatures (20, 40, and 60[degrees]C) and three low (and therefore isothermal) constant displacement rates: 4 X [10.sup.-6], 2 X [10.sup.-5], and 1 X [10.sup.-4] m/s.

Since it was not possible to predict where necking would begin, the gauge section length of each specimen was marked off into four intervals. The draw ratio was determined both from the extension and from the reduction in cross-sectional area after one of these intervals had completely drawn. It was found that careful specimen molding and machining was needed to achieve a sufficiently uniform length of fully and steadily drawn polymer.

The 12-mm plaques were used to manufacture Reversed Charpy specimens, 12 mm wide and 70 mm long. These were razor-notched to leave a ligament s = 1.5 mm and tested at 0[degrees]C, with an impact speed of 1 m/s on a span of 60 mm, using the procedure developed by Hillmansen (7). The load-displacement record was integrated to determine the absorbed energy U and hence to determine [w.sub.p]:

[w.sub.p] = [U/[B[s.sup.2]]], (5)

where B is the notch width.


Figure 5 shows Eyring plots for [G.sub.p] and [[sigma].sub.y] in one of the seven materials (Material Pipe 3 from Ref. 15). The six Eyring parameters were determined by a multivariate regression method, using Microsoft [Excel.sup.R], from nine groups of data points (each group containing three repeat test results). Although the range of strain rate is small, the consistency in behavior across the temperature range was good enough for the data to be extrapolated with confidence. The critical thermomechanical conversion factor [[beta].sub.c] was determined by iteration, involving an adiabatic simulation at each successive [beta] value. As noted previously both in experiments (6) and in simulations (10), drawing above [[beta].sub.c] became unstable at a temperature of 50-60[degrees]C, i.e. within the range for which isothermal data were measured. Figure 6 shows an example of adiabatic stress/strain curves integrated from the Haward-Thackray-Eyring material model.



Figure 7 shows that despite experimental and analytical uncertainties [[beta].sub.c] provides a very effective index for the plane-stress fracture resistance [w.sub.p]. Figure 8a and b illustrate the robustness of the procedure in that neither the initial temperature of the simulation nor the strain rate chosen to characterize plane-stress separation unduly influences the result.




The results presented here provide a signpost for the development of PEs with low RCP critical temperatures. The Eyring model is sufficiently physical to encourage interpretation of its parameters in microstructural terms, and simple tensile testing allows each interpretation to be tested for well-characterized PEs. However, it is important to emphasize three specific limitations.

First, the use of a "critical value" of [beta] as an index of quality rests on the reasonable assumption that adiabatic drawing stability is determined by ([[beta].sub.c] - [beta]), and the more questionable assumption that [beta] [approximately equal to] const. The actual values of thermomechanical efficiency [beta] for these polye-thylenes must certainly lie in a more limited range than that of the computed [[beta].sub.c] values--since some [[beta].sub.c] values exceed unity, and [beta] cannot. However, [beta] may depend on crystallinity, which will decrease with increasing cooling rate. There is some evidence for this effect in the data of Fig. 8, and scatter is substantially lower within each constant cooling rate data subset than it is for the set as a whole.

Second, some uncertainty is involved in the extrapolation of isothermal data over such a large range of strain rate, as Fig. 5 emphasizes. The range of strain rates available for testing is constrained: it is important to obtain a representative length (at least 20 mm) of drawn polymer in a reasonable time, but the strain rate in the neck "shoulder" should not exceed [10.sup.-2] [s.sup.-1] to maintain is othermal conditions. The Eyring model does allow a rate sensitivity fit to be obtained using all the available data points from several test temperatures, and the rate sensitivity does seem to be well correlated. Figure 5a suggests that the temperature sensitivity is less well represented, and this may reflect the onset of a new yield process between 40 and 60[degrees]C. However, no extrapolation is needed beyond the range of temperatures tested, which is sufficiently well represented.

Third, [w.sub.p] alone cannot be expected to provide a performance index for RCP in pipe. The requirement is for a high "critical pressure," below which RCP arrests and above which it continues indefinitely. Although fracture models relating critical pressure for gas-pressurized pipe to effective fracture toughness exist, they are complicated and nontransparent. Nevertheless, it can be taken for granted that critical pressure depends on dynamic fracture resistance.

The effective dynamic fracture resistance of a total width B of crack front with a plane-stress margin of width [w.sub.p] on each side is

[G.sub.D] = (1 - [2s/B])[G.sub.ID] + 2[[s.sup.2]/B][W.sub.p], (6)

where [G.sub.ID] is the plane strain dynamic fracture resistance. We have previously used two different methods to measure [G.sub.ID]. The High-Speed Double Torsion test (16) uses a large (100 x 200 mm) thick plaque specimen, and allows the crack speed to be varied. The Hydrostatic S4 test (17), uses a water-pressurized pipe specimen, and allows the crack to propagate only at the speed for which [G.sub.ID] is minimum. In both methods, that part of the surface which would separate by plane stress must first be removed by a sharp "sidegroove." Values for PE are typically low, as expected (2-3 kJ/[m.sup.2]) and appear not to be very temperature-sensitive. [G.sub.ID] can also be predicted using the adia-batic decohesion fracture model (18)

[G.sub.ID] = 5[rho][[bar.s].sub.W][[C.sub.p]([T.sub.m] - T) + [DELTA][H.sub.f]], (7)

are Greenshields and Leevers (17) found that this simple formula worked well for a number of thermoplastics.

Now the plane-stress layer thickness s must be predicted. Williams (3) suggested that s = [r.sub.p], where [r.sub.p] is the plane-stress plastic zone size given by

[r.sub.p] = [1/[2[pi]]][[EG.sub.ID]/[[sigma].sub.y.sup.2]], (8)

and E is the tensile modulus. Equations 6-8 begin to mark out a scheme by which the influence of material properties on critical pressure can be understood: [G.sub.ID] contributes directly, but also determines the width of the (usually) small plane-stress region within which for typical total thicknesses of a few millimeters--[w.sub.p] makes a much greater contribution.


Plane-stress fracture resistance at high rates is a key property for RCP resistance in polyethylenes destined for pressure pipe applications. Seven materials of widely differing structure and plane-stress fracture resistance are correctly ranked using an index of tensile drawing stability under adiabatic conditions. The basic nature of the material properties used to calculate this stability index (the Eyring parameters of yield and strain hardening processes) suggests that the tensile testing procedure could provide a valuable predictive understanding of what structural parameters make a good, RCP-resistant, pipe-grade PE.


The authors express their sincere appreciation to Chevron Phillips Chemical, Bartlesville, OK for the material and technical support provided to Pemra Ozbek during this work.


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Pemra Ozbek, Patrick Leevers

Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK

Correspondence to: Patrick Leevers; e-mail:

Contract grant sponsors: Chevron Phillips Chemical Company LLC; Universities UK (Overseas Research Student award).
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Author:Ozbek, Pemra; Leevers, Patrick
Publication:Polymer Engineering and Science
Article Type:Technical report
Geographic Code:1USA
Date:Sep 1, 2008
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