Application of essential work of fracture concept to toughness characterization of high-density polyethylene.

INTRODUCTION

The essential work of fracture (EWF) [1-5] is a concept for material toughness evaluation. Its value represents the energy consumed within the fracture process zone (FPZ) where new surface is generated. For crack growth subjected to tensile loading, the EWF value is often determined using double-edge-notched tensile (DENT) test [6], for which the specimen is shown in Fig. 1. Under the plane-stress condition, total work consumed in a DENT test consists of two parts: (i) the essential energy for the formation of new fracture surface, and (ii) the energy for plastic deformation around the ligament section. The EWF concept is a scheme to extract the energy for part (i) from the total fracture energy in the DENT test, and is increasingly popular for evaluation of the toughness for ductile fracture [6].

For materials in ductile fracture, FPZ is known to undergo a necking process that eventually breaks down to form fracture surface. It has been suggested that for necking to be fully developed in the FPZ, specimen dimensions have to meet the following constraints [3, 7]. The EWF value so determined is known as the plane-stress specific EWF ([w.sub.e]).

(3 - 5)[t.sub.0] [less than or equal to] [L.sub.0] [less than or equal to] min ([W/3] or 2[r.sub.p]) (1)

where [t.sub.0], [L.sub.0] and W are specimen thickness, ligament length and specimen width, respectively, and 2[r.sub.p] is the size of the plastic zone that can be estimated using the following equation:

2[r.sub.p] = [1/[pi]][E[w.sub.e]/[[sigma].sub.y.sup.2]] (2)

where E is the elastic modulus and [[sigma].sub.y] the tensile yield stress. Value of [w.sub.e] represents energy consumed for the ormation of neck and new fracture surface inside the FPZ. Some work [3, 8-10] has attempted to express the EWF value in terms of the energy consumed before and after the neck formation in FPZ, to take into account the energy for the necking explicitly.

There are two types of necking behaviour, based on the stability of the necking process [11]. For many ductile materials, the neck development is an unstable process that leads to final fracture soon after the neck is initiated. This is because increase of strength by the work-hardening from the neck formation cannot keep up with the stress increase caused by the cross section reduction. However, for some very ductile materials like high-density polyethylene (HDPE), the work-hardening rate in the neck is fast enough to compensate for the decrease of the cross-sectional area, so that the neck development process is stabilized, and its size grows at a constant rate [12].

In a plane-stress DENT test of HDPE, neck is initially initiated along the ligament section. As to be shown in this article, after the neck is formed through the whole ligament, it propagates into the neighboring region that has been plastically deformed at the early stage of the test. In the past, studies that evaluated [w.sub.e] of HDPE [3, 8, 13] have largely ignored the neck development process for the energy consumption analysis.

Objective of this study is to examine the deformation behaviour of DENT specimens of HDPE in the plane-stress condition, to identify the deformation mechanisms involved and to quantify the associated energy consumption. Uniaxial tensile (UT) tests were also conducted to extract the stress-strain relationship of HDPE during the necking process. The information is used to facilitate the understanding of the neck development process.

[FIGURE 1 OMITTED]

This article will firstly present results from the UT tests, for analysis of the stress-strain relationship during the necking process. With the assumption that the stress-strain relationship from the UT test is applicable to the DENT test, deformation process involved in the latter is examined, the associated energy consumption at each stage of the crack growth quantified, and their specific EWF values determined. The specific EWF values are then compared with those determined from the conventional method that uses total energy consumption for the analysis.

EXPERIMENTAL DETAILS

Specimens used in the study were machined from a commercial-grade, extruded HDPE plate of 6.25-mm-thick and density of 0.96 g/[cm.sup.3], provided by McMaster-Carr, USA. The HDPE is same as that used in the previous study [14]. To ensure consistency of material properties for the DENT and the UT specimens, both types of specimens were tested with length along the rolling and the transverse directions. However, results from the UT tests showed that the specimen orientation has little effect on the stress-strain curves. Therefore, the UT test results presented in the article are only from specimens in the transverse direction.

Preliminary UT tests confirmed that the width-to-thickness ratio of a given cross section remains constant during the plastic deformation and the necking process. Therefore, strain generated in the UT tests was determined by the width change, measured using an extensometer mounted across the specimen width in a section where the necking occurred. The nominal strain ([[epsilon].sub.E]) and the true strain ([epsilon]) generated in the UT tests were determined using the following equations:

[[epsilon].sub.E] = ([W.sub.0]/W)[.sup.2] - 1 (3)

[epsilon] = 2 ln ([W.sub.0]/W) (4)

where [W.sub.0] is the original specimen width and W the specimen width during the test.

The UT tests were conducted at a crosshead speed of 5 mm/min, using a QUASAR 100 test frame that has a load capacity of 100 kN. To ensure that necking occurred in the gauge section where the extensometer was mounted, width around that section was reduced by 0.1 mm (1% of the width). Such width reduction is within the limit recommended in the ASTM standard for the UT test [15]. It should be noted that the final fracture did not occur in the section where the width was reduced, supporting that the width reduction did not affect the test results.

In the DENT tests, the transition from the plane-stress fracture to the mixed plane-stress/plane-strain fracture could easily be detected by the deviation of the linear trend line, in the plot of specific work of fracture versus ligament length. It has been suggested that the lower bound of the initial ligament length for the plane-stress fracture should be three times of the initial ligament thickness ([L.sub.0] = 3[t.sub.0]) [3, 8, 13]; however, our previous study [14] has shown that for HDPE, the plane-stress fracture could occur with [L.sub.0] down to 2.4[t.sub.0]. Therefore, the DENT specimens used in this study, Fig. 1, were designed to have width (W) 90 mm and the initial ligament length ([L.sub.0]) ranging from 15 to 32 mm (corresponding to 2.4 and 5.12 times of [t.sub.0]). Overall length of the specimens was 260 mm. Because the plane-stress specific EWF value ([w.sub.e]) of the HDPE depends on the specimen orientation [14], DENT tests were conducted on both types of specimens that have length along either the rolling or the transverse direction. The tests were conducted using an Instron testing frame that has a load capacity of 250 kN, at a crosshead speed of 5 mm/min.

The strain in the ligament section of the DENT specimens was also determined using the change of the cross sectional dimensions. Following the assumption that the strain of the DENT specimens in the ligament length direction is negligible [16], the true strain in the loading direction in the ligament section was determined using the following equation:

[epsilon] = ln([t.sub.0]/t) (5)

where [t.sub.0] is the original specimen thickness and t the thickness during the test.

[FIGURE 2 OMITTED]

STRESS-STRAIN RELATIONSHIP FROM THE UT TEST

A typical time function of the loading curve from the UT tests is presented in Fig. 2a. The corresponding [sigma] - [epsilon] curve is shown in Fig. 2b, and [[sigma].sub.E] - [[epsilon].sub.E] curve in Fig. 2c. Necking started at the maximum load where [epsilon] was around 0.12 (the equivalent [[epsilon].sub.E] was about 0.127). As expected, increase of the strain after the onset of necking was mainly caused by the reduction of the cross section. The term "neck inception stage" will be used here to refer to the deformation process, in which reduction of the cross sectional area was localized for the neck formation. After the load passed through the minimum point, the neck started growing into the neighboring region, and the total length of the neck increased significantly. This part of the necking process will be referred to as the "neck propagation stage," in which the load never decreased again until the specimen fractured. Although the load increased, decreased, and then increased again in the UT tests, the true stress-strain curve, Fig. 2b, shows monotonic increase of the true stress without any drop, at both the neck inception and the neck propagation stages.

Figure 2b also contains a curve that was generated numerically to fit the experimental data in the plastic deformation regime. The curve in the low-strain section, i.e., below [[epsilon].sub.0] in Fig. 2b, was generated based on a constitutive equation that was originally proposed by Hollomon [17]:

[sigma]([epsilon]) = K[[epsilon].sup.N] (6)

where K is the strength coefficient and N the work-hardening coefficient. The section above [[epsilon].sub.0] in the curve was generated using an expression proposed by G'Sell and Jonas [12]:

[sigma]([epsilon], [dot.[epsilon]]) = k exp(M[[epsilon].sup.[beta]]) [dot.[epsilon].sup.m] (7)

where k, M, and m are material constants and [dot.[epsilon]] the strain rate. With the assumption that the true stress-strain curve is insensitive to the strain rate, the earlier expression can be rewritten as:

[sigma]([epsilon]) = k' exp(M[[epsilon].sup.[beta]]) (8)

where k' can also be regarded as a material constant.

Hill [16] proposed that the necking occurs when the following condition is satisfied:

[1/Y][dY/d[epsilon]] [less than or equal to] ([[partial derivative]f/[[partial derivative][[sigma].sub.1]]] + [[partial derivative]f/[[partial derivative][[sigma].sub.2]]])/[[partial derivative]f(Y,0)/[partial derivative]Y] (9)

where Y is uniaxial yield strength of the material, f a yield function, and [[sigma].sub.1] and [[sigma].sub.2] the principle stresses. By choosing von Mises function as f, that is,

f [equivalent to] [[sigma].sub.1.sup.2] - [[sigma].sub.1][[sigma].sub.2] + [[sigma].sub.2.sup.2] = [Y.sup.2] (10)

it can be shown that based on Eq. 9, necking occurs under uniaxial loading when

[1/Y][dY/d[epsilon]] [less than or equal to] 1. (11)

A strain range for the necking in the UT test can be determined by combining Eq. 11 with Eqs. 6 and 8, which is

N [less than or equal to] [epsilon] [less than or equal to] (1/M[beta])[.sup.(1/[[beta]-1])]. (12)

However, the earlier strain range is not directly applicable to the necking process in the DENT test. Following the assumption by Hill [16] that the plastic strain along the ligament length direction remains constant during the DENT test, with the use of von Mises function as f in Eq. 9 the necking in the test is expected to occur when the following condition is met:

[1/Y][dY/d[epsilon]] [less than or equal to] [square root of 3]/2. (13)

The corresponding strain range for the necking in the DENT test is:

N/[[square root of 3]/2] [less than or equal to] [epsilon] [less than or equal to] ([[square root of 3]/2]/M[beta])[.sup.(1/[[beta] - 1])]. (14)

In this study, data from the UT tests were used to determine the constants in Eqs. 6 and 8, with the value of N (equivalent to the strain for the onset of necking in the UT tests) being 0.12. Value of K in Eq. 6 was determined to be 37.5 MPa for the best curve fit in the plastic strain range, which is for strain in Fig. 2b to be up to [[epsilon].sub.0] (0.32).

Values of M and k' were determined by imposing continuity of the stress and the tangent modulus (d[sigma]/d[epsilon]) between Eqs. 6 and 8 at the strain 0.32, which yielded the following expressions for M and k' in terms of [beta]:

M = N/[[beta] [[epsilon].sub.0.sup.[beta]]] (15)

k' = [K [[epsilon].sub.0.sup.N]]/[exp(N/[beta])]. (16)

Through trial and error, the best curve that fits data in Fig. 2b in the strain range above [[epsilon].sub.0] was generated by Eq. 8 based on [beta] value of 1.6. The corresponding k' and M values are 30.4 and 0.47, respectively. Note that the [beta] value is close to 2 that was suggested by G'Sell and Jonas [12] for HDPE.

Using the earlier N, M, and [beta] values, Eq. 12 estimates that the true strain range for HDPE to develop necking in the UT test is between 0.12 and 1.61, equivalent to the nominal strain range from 0.13 to 3.99, as illustrated by the shaded area in Fig. 2c. The corresponding loading range is presented in Fig. 2a in the section that is also high-lighted by the shaded area.

As predicted by Eq. 12, the maximum strain for necking in the UT test is equivalent to 55% reduction of the specimen width, defined as (1 - [W/[W.sub.0]]). The width reduction in the fractured UT specimens was examined and found to vary from 58% at the centre of the neck to 50% at the end of the neck. Since these specimens are already fractured, the measured values should have excluded the width reduction due to the elastic deformation, of about 5%. Therefore, the upper limit predicted by Eq. 12, 55%, can serve as a conservative estimate of the maximum strain that can possibly be generated by the necking process in the UT tests. Note that the nominal strain in Fig. 2c was measured from the position where the neck was initiated. After the neck propagation commenced, cross section in the already necked region was reduced continuously but at a much slower rate, as shown by the dense data points at the very end of Fig. 2c.

At the molecular level, the neck development is attributed to the uncoiling and alignment of polymer chain segments [18]. When tensile stress on the specimen increases to a critical level, the entangled molecular chains start to uncoil at a particular section, which corresponds to the "neck inception stage." The uncoiling and alignment continues at the "neck propagation stage." It has been reported that at an elevated temperature HDPE can be drawn to a draw ratio [LAMBDA] (defined as the ratio of the final length to the initial length) between 8 and 10 [19]. However, at room temperature the maximum A of HDPE is about 6 that is equivalent to an engineering strain ([[epsilon].sub.E]) of 500% [20]. When the elongation approaches this level, the specimen becomes noticeably more rigid. These phenomena are consistent with those shown in Fig. 2b and c, in which a sharp rise of the stress occurs in the right-most region of [[epsilon].sub.E], of around 550% which corresponds to A of about 6.5. When a sufficient strain is generated in the neck, fracture occurs.

DEFORMATION BEHAVIOUR IN THE DENT TEST

Figure 3a demonstrates the typical shape of load-displacement curves from DENT tests in the plane-stress condition. The curves always contain a transition of the load drop rate which is indicated by point B on the top curve. Figure 3b presents all values of specific work of fracture ([w.sub.f], defined as the total work divided by the area in the ligament section) in the plane-stress condition, plotted as a function of the ligament length [L.sub.0]. Through linear regression to zero ligament length, Fig. 3b suggests that specific EWF values ([w.sub.e]) in the plane-stress fracture for specimens in the transverse and the rolling directions are 49.7 and 75.5 kJ/[m.sup.2], respectively. These values are comparable to values published in the literature, in the range from 35 kJ/[m.sup.2] [13] to 78 kJ/[m.sup.2] [3, 8], though no specific specimen orientation was provided in the literature.

The earlier data deduction process has been widely adopted for the DENT test, which is based on the following assumptions for determining [w.sub.e]: (i) the plastic zone is fully developed around the ligament section prior to the crack growth, (ii) the necking is confined within the FPZ, and (iii) the height of FPZ and fracture strain within the FPZ remain constant during the crack growth [3, 5, 21]. The latter two assumptions imply that work hardening in the material should not be significant enough to generate stable necking. However, the UT tests suggest that the work hardening in the HDPE is strong enough to stabilize the neck development process, to allow the neck propagation before the specimen fractures. This raises concerns on whether the earlier data deduction process is applicable to the HDPE.

[FIGURE 3 OMITTED]

It is worth mentioning that in Broberg's original paper that introduced the EWF concept [1], fracture is categorized in three main classes based on the characteristics of the end region (FPZ): (i) with an autonomous unstable end-region in brittle materials, (ii) with an unstable end-region in ductile materials, of which dimensions are proportional to the crack length, and (iii) with a stable end-region in very ductile materials. Since the EWF concept is based on the assumption of unstable necking in FPZ, it is applicable to materials in the first class. The concept may also be applicable to materials in the second class, provided that fracture in FPZ is autonomous and fracture criteria for the first class are applicable to their crack growth process. However, the assumption of constant height of FPZ during the crack growth is not applicable to materials in the third class such as HDPE that generates stable FPZ. As a result, validity of the earlier data analysis that is based on the EWF concept, but without the consideration of its limitation, needs to be examined.

The curves from the DENT tests, Fig. 3a, clearly indicate that the HDPE specimens used in this study fractured in two stages, between which transition caused the change of the load drop rate. Deformation in the two stages is demonstrated in Fig. 4. The top photograph in the figure was taken above the fracture surface (top view); the bottom left from the specimen surface (front view); and the bottom right along the ligament length direction (side view). The front view clearly shows two stages of deformation during the crack growth. The first stage, marked [L.sub.i]/2 for the left half in the front view, was initiated soon after the maximum load was reached, i.e., point A in Fig. 3a. As to be explained later, neck development at this stage is equivalent to the "neck inception stage" in the UT test. The second stage, marked [L.sub.p] in the front view of Fig. 4, was initiated after the change of the load drop rate, as indicated by point B in Fig. 3a. Neck development at this stage is equivalent to the "neck propagation stage" in the UT test.

Thickness decrease in the ligament section of the DENT specimens was examined after the test. Typical thickness change is shown in Fig. 4, which was taken from a specimen with [L.sub.0] = 27 mm. Top view of Fig. 4 suggests that at the neck inception stage the ligament thickness decreases continuously with the crack growth. The strain range that corresponds to the thickness decrease at this stage was found to be within that specified by Eq. 14. At the neck propagation stage, i.e., after the necking has been initiated through the entire ligament section, strain in the ligament section only increased slightly with the crack growth. The ligament thickness maintained almost constant of around 1.6 mm (from the initial thickness of 6.25 mm). With the assumption of zero strain in the ligament length direction, as suggested in Ref. 16, the strain in the loading direction, based on the specimen thickness reduction, was found to be around 320%. This value is far below the strain in the very right region of the stress-strain curve from the UT test, as shown in Fig. 2c where the stress increases sharply.

The deformation and fracture process in the DENT test is schematically presented in Fig. 5. The top two drawings depict the development of the plastic zone, the FPZ and the onset of crack growth when the maximum load is reached, i.e., point A in Fig. 3a. Note that the drawings suggest that the plastic zone in these specimens has a rather flat elliptical shape, in contrast to the nearly circular shape for metallic specimens. Figure 5 also suggests that the crack growth is accompanied with the increase of the FPZ size, with the size increase in the ligament length direction being faster than that in the specimen length direction. Since the length of FPZ increased faster than the crack growth, the two FPZs met at the centre of the specimen, as illustrated by the third drawing in Fig. 5, with the crack tips still at a distance [L.sub.p] away from each other. The corresponding point on the load-displacement curve is point B in Fig. 3a, known as the transition point. Further specimen elongation caused stable neck propagation in the loading direction, while the two crack tips proceeded steadily towards the centre. The fracture process after the transition point resulted in a necked zone of a triangular shape, as shown by the bottom drawing in Fig. 5.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Figure 6 shows an example of the DENT specimens at the transition point, at which the neck has developed across the whole ligament section. This necking process bears some similarity with that in the UT test. The main difference is that in the DENT tests, the crack growth was involved in both the neck inception and the neck propagation stages, while in the UT tests, the crack growth occurred only at the end of the neck propagation stage.

Interestingly, crack growth speed in the DENT tests remained nearly constant during both stages of the neck development. This is evident in Fig. 7, which resents the reduction of the ligament length as a function of time for two specimens of the same [L.sub.0], with time zero being the point when the maximum load is reached. The constant crack growth speed allowed us to use the load-displacement curve to determine the remaining ligament length at any point during the DENT test. That is, based on values of [L.sub.0] and displacement at the point of interest, [DELTA], we can determine the ligament length L between the two crack tips using the following expression:

L = [[[DELTA] - [[DELTA].sub.y]]/[[[DELTA].sub.f] - [[DELTA].sub.y]]][L.sub.0] (17)

where [[DELTA].sub.y] is the displacement at the maximum load, and [[DELTA].sub.f] the displacement at fracture. Eq. 17 suggests that lengths of the ligament sections for the two stages of crack growth, [L.sub.i] and [L.sub.p], can be determined using the following expressions:

[L.sub.i] = [[[[DELTA].sub.n] - [[DELTA].sub.y]]/[[[DELTA].sub.f] - [[DELTA].sub.y]]][L.sub.0]; [L.sub.p] = [[[[DELTA].sub.f] - [[DELTA].sub.n]]/[[[DELTA].sub.f] - [[DELTA].sub.y]]][L.sub.0] (18)

where [[DELTA].sub.n] is the displacement at the transition point.

Experimental results suggest that values of [[DELTA].sub.y]/[[DELTA].sub.f] and [[DELTA].sub.n]/[[DELTA].sub.f] are nearly constant in the whole range of ligament lengths used in the study, as shown in Fig. 8. Mean value of [[DELTA].sub.y]/[[DELTA].sub.f] is 0.20 for both rolling and transverse directions, and that of [[DELTA].sub.n]/[[DELTA].sub.f] is 0.39 and 0.42, respectively. Since these values are independent of [L.sub.0], the ratios of [L.sub.i] to [L.sub.0] and [L.sub.p] to [L.sub.0] should also be independent of [L.sub.0]. The latter ratio for the HDPE used in this study is 0.75 and 0.72, respectively, in the rolling and the transverse directions.

Neck Propagation in DENT Test

For polymers with limited ductility, the plastic deformation does not generate sufficient work hardening to stabilize the neck development process. Their DENT specimens are expected to fracture at the neck inception stage. For polymers with high ductility such as HDPE in the plane-stress condition, significant work hardening slows down the crack formation and allows the occurrence of neck propagation before the final fracture.

[FIGURE 7 OMITTED]

In addition to the earlier mechanism that is based on the different stability in the necking process, the plastic zone shape is also believed to attribute to the occurrence of neck propagation in the DENT test of highly ductile polymers. The explanation is as follows. Strain increment (d[epsilon]) in the active plastic zone can be expressed in terms of displacement increment (du), the remaining ligament length (L), and the active plastic zone shape (represented by the shape factor [lambda]) [5]:

d[epsilon] = du/[lambda]L. (19)

Since the cross-head speed (du/dt) is constant during the test, the strain rate (d[epsilon]/dt) in the FPZ of a given L depends only on the [lambda] value. For metallic materials, shape of the plastic zone is nearly circular, thus [lambda] being close to 1. But for polymeric materials like HDPE, because of the flat shape of the plastic zone, [lambda] is much smaller than 1. Therefore, the strain rate in the ligament section of the polymeric specimens should be much higher than that in the metallic specimens when subjected to the same testing conditions, allowing the former to reach the strain for neck propagation before the crack tips reach the centre of the ligament. As a result, the central section of the ligament for polymeric specimens could go through the neck propagation stage before the specimen fractured completely.

Transition of the Load Drop Rate in the DENT Test

Another interesting phenomenon in the DENT test is the transition of the load drop rate, i.e., point B in Fig. 3a. This phenomenon can be explained based on the variation of the average nominal stress ([[sigma].sub.E]) between the neck inception stage and the neck propagation stage. Let F be the load carried by the remaining ligament section during the DENT test. F can be expressed as:

F = [[sigma].sub.E] [t.sub.0] L. (20)

The load drop rate (dF/dt) during the crack growth is:

dF/dt = [d[[sigma].sub.E]/dt][t.sub.0] L + [[sigma].sub.E][t.sub.0][dL/dt]. (21)

With a constant crack growth speed through the test, i.e., a fixed, negative value of dL/dt, difference of dF/dt between the two stages of neck development is dominated by the difference of the first term on the right-hand side of Eq. 21. Results from the UT tests, Fig. 2c, suggest that [d[[sigma].sub.E]]/dt should be negative in the neck inception stage, but zero or slightly positive in the neck propagation stage. Therefore, the load drop rate (the absolute value of dF/dt) in the neck inception stage should be higher than that in the neck propagation stage.

Fracture Criterion

The deformation behaviour observed in the DENT tests clearly suggests that critical elongation, though being a common fracture criterion for ductile materials, is not suitable for the FPZ fracture in HDPE. Chen et al. [22] proposed an alternative criterion for the crack growth in HDPE that is based on the crack-tip opening angle (CTOA). Since the load-displacement curves of their HDPE specimens did not show any existence of a neck propagation stage, suitability of CTOA as a criterion for the HDPE used in this study needs to be re-evaluated.

Here, the crack tip angle was measured in situ in both the neck inception and the neck propagation stages, as presented in Fig. 9a and b, respectively. CTOA for 10 DENT specimens were monitored, and were found to maintain around 130[degrees] through the entire necking process, supporting that the CTOA is a valid criterion for the crack growth in the HDPE used in this study, even in the neck propagation stage.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Maintaining a constant crack tip angle in the DENT test can also be rationalized in the following way. As shown in Fig. 9c, angle at the crack tip ([alpha]), specimen elongation ([DELTA]), and ligament length (L) have the following relationship:

d[DELTA]/dL = - tan([alpha]/2). (22)

Since the DENT tests were conducted at a constant cross-head speed (d[DELTA]/dt) and the crack growth rate (dL/dt) maintained constant during the test, d[DELTA]/dt should be constant, resulting in a constant [alpha] value.

ENERGY CONSUMPTION IN THE DENT TEST AND THE EWF VALUES

In view of the significant difference in the deformation behaviour between the neck inception and the neck propagation stages in the DENT test, toughness of HDPE determined from the test may need more than single [w.sub.e] to characterize. Instead of using the total energy consumption, a work-partitioning scheme is proposed here to isolate the energy consumption for each stage of the crack growth to determine the corresponding [w.sub.e] values.

Several work partitioning schemes have been proposed in the past for various purposes. The first scheme proposed for polyethylene was by Mai et al. [3, 8] who divided the total work of fracture into the work for fracture initiation ([w.sub.i]) and the work for fracture ([w.sub.f]), as shown in Fig. 10a. The corresponding specific work for fracture initiation ([w.sub.i]) is determined by linear regression of [W.sub.i]/[[L.sub.0][t.sub.0]] to zero ligament length, which is suggested to represent some kind of material toughness that is less than [w.sub.e], but greater than the plane-strain specific EWF. Later, Karger-Kocsis and coworkers [4, 23, 24] proposed a work partitioning scheme to determine the work for yielding ([W.sub.y]) and the work for the subsequent necking and tearing ([W.sub.n]), as shown in Fig. 10b. Through normalization by the area of the original ligament section ([L.sub.0][t.sub.0]), based on the linear regression to zero ligament length, the specific work of fracture ([w.sub.f]) is expressed as:

[w.sub.f] = [w.sub.f,y] + [w.sub.f,n] = ([w.sub.e,y] + [[beta].sub.y][w.sub.p,y][L.sub.0]) + ([w.sub.e,n] + [[beta].sub.p][w.sub.p,n][L.sub.0]) (23)

where [w.sub.e,y], [w.sub.e,n], [w.sub.p,y] and [w.sub.p,n] are specific EWF for yielding, specific EWF for necking, specific work for plastic deformation in yielding, and specific work for plastic deformation in necking, respectively. They proposed that [w.sub.e,y] is the critical EWF for plane-strain fracture that represents generic toughness of the material [25, 26]. However, little evidence was provided to support the claim.

Based on our experimental observation of the HDPE deformation in the plane-stress DENT test, the total work of fracture should be divided into two parts, to determine the EWF values for the two stages of the crack growth. The division is depicted in Fig. 10c, by a vertical line at point B where the transition of the necking process occurs. With the assumption that the work for elastic deformation is negligible in comparison with the total work consumed for fracture, the work for each stage of crack growth is represented by the area under the corresponding load-displacement curve, that is, [W.sub.i] for the neck inception stage and [W.sub.p] for the neck propagation stage. Crack growth length in each stage is [L.sub.i] and [L.sub.p], which can be determined using Eq. 18.

The work of fracture is normalized by the original ligament area for the corresponding stage of crack growth. The expressions for the specific work of fracture for two stages of crack growth are:

[w.sub.f,i] = [W.sub.i]/[L.sub.i][t.sub.0] and [w.sub.f,p] = [W.sub.p]/[L.sub.p][t.sub.0]. (24)

Similar to the approach proposed by Karger-Kocsis et al., each specific work of fracture is believed to contain an essential term that is independent of the ligament length and a nonessential term that is proportional to the ligament length. That is,

[FIGURE 10 OMITTED]

[w.sub.f,j] = [w.sub.e,j] + [[beta].sub.j][w.sub.p,j][L.sub.i]

[w.sub.f,p] = [w.sub.e,p] + [[beta].sub.p][w.sub.p,p][L.sub.p]. (25)

The earlier equations can also be expressed in terms of [eta] (defined as [L.sub.p]/[L.sub.0]) as:

[w.sub.f,i] = [w.sub.e,i] + [[beta].sub.i][w.sub.p,i](1 - [eta])[L.sub.0]

[w.sub.f,p] = [w.sub.e,p] + [[beta].sub.p][w.sub.p,p][eta] [L.sub.0]. (26)

Using linear regression to zero ligament length, [w.sub.e,i] and-[w.sub.e,p] for the HDPE in the rolling direction are 201.5 and 44.3 kJ/[m.sup.2], respectively, as shown in Fig. 11a. The corresponding values in the transverse direction are 144.0 and 21.7 kJ/[m.sup.2], Fig. 11b. These values suggest that [w.sub.e] for the HDPE is not constant during the crack growth in the DENT test, but varies with the deformation mechanisms involved in the crack growth. Value of [w.sub.e] measured by the conventional method can be regarded as an average value for the fracture toughness that may actually vary during the crack growth.

It should be noted that Eq. 26 can be used to determine [w.sub.e,i] and [w.sub.e,p] only because the [eta] values are insensitive to the change of the ligament length, as shown in Fig. 12. Otherwise, values of [w.sub.e,i] and [w.sub.e,p] have to be determined using Eq. 25.

The earlier [w.sub.e,i] and [w.sub.e,p] values did not take into account the thickness change during the crack growth, thus the values may have underestimated the fracture toughness of the HDPE. If the thickness change had been considered in the earlier calculation, values of [w.sub.e,i] and [w.sub.e,p] would have been bigger, with the latter being more than three times of the earlier values due to the thickness decrease from 6.25 mm to about 1.6 mm. That is, values of [w.sub.e,p] would have been 168 and 83 kJ/[m.sup.2] for specimens in the rolling and transverse directions, respectively. Nevertheless, the earlier analysis suggests that the specific EWF values for HDPE are not constant. Further study will be conducted to elucidate the variation of [w.sub.e] during the DENT test, especially with the consideration of the change of the specimen thickness.

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

DISCUSSION

The difference between [w.sub.e,i] and [w.sub.e,p] could have been caused by the difference of the deformation mechanisms involved in the crack growth. Before the transition of the load drop rate (prior to point B in Fig. 3a), the deformation mechanisms included the neck inception across the whole ligament section and the neck propagation within the distance of [L.sub.i]/2 from the notch tips. After point B, the deformation was dominated by the neck propagation in the active plastic zone.

Mai and coworkers [3, 8-10] suggested that the energy consumed within the FPZ can be expressed as the sum of two terms, one for the plastic deformation before the onset of necking and the other for the neck formation and fracture. Based on this concept, Mai and coworkers expressed [w.sub.e] as [3, 8-10]:

[w.sub.e] = h [[integral].sub.0.sup.[bar.[epsilon].sub.n]] [bar.[sigma]]d[bar.[epsilon]] + [[integral].sub.[[epsilon].sub.n]h.sup.[[DELTA].sub.F]] [sigma]([[DELTA].sub.1])d[[DELTA].sub.1] (27)

where h is the height of the FPZ and is assumed to be constant during the test, [bar.[sigma]] and [bar.[epsilon]] are the true stress and true strain in the FPZ, [bar.[epsilon].sub.n], and [[epsilon].sub.n] are the true strain and engineering strain, respectively, at the onset of necking, [sigma] is the engineering stress during the necking process, which is a function of the crack tip opening displacement ([[DELTA].sub.1]), and [[DELTA].sub.F] is the displacement [[DELTA].sub.1] at fracture. However, Eq. 27 can be applied only if unstable necking occurs within the FPZ. In the case that the deformation transition occurs during the crack growth, contribution from the two terms in Eq. 27 to the [w.sub.e] value may vary, due to the change of the deformation mechanisms.

To investigate the possible variation of the two terms in Eq. 27 for contributing to the [w.sub.e], value for the crack growth with stable necking is divided into two terms, [w.sub.e,1] that represents the specific EWF for the neck inception within the FPZ and [w.sub.e,2] for the specific EWF for the neck propagation in FPZ to the final fracture. With the assumption that strain within FPZ is constant, [w.sub.e,1] and [w.sub.e,2] can be expressed as:

[w.sub.e,1] = [h.sub.1] [[integral].sub.0.sup.[bar.[epsilon].sub.n,max]] [bar.[sigma]]d[bar.[epsilon]] (28)

[w.sub.e,2] = [h.sub.2] [[integral].sub.[bar.[epsilon].sub.n,max].sup.[bar.[epsilon].sub.f]] [bar.[sigma]]d[bar.[epsilon]] (29)

where [bar.[epsilon].sub.n,max] is the maximum equivalent true strain for the neck inception in the FPZ, [bar.[epsilon].sub.f] the equivalent true strain at the final fracture.

The total EWF consumed in the neck inception stage ([w.sub.e,i]) of the DENT test should be the summation of the work for the neck inception through the entire ligament section and the work for the neck propagation within the distance [L.sub.i]/2 from each notch tip. That is,

[W.sub.e,i] = 2 [t.sub.0][[[integral].sub.0.sup.[L.sub.0]/2] [w.sub.e,1]dL + [[integral].sub.0.sup.[L.sub.i]/2] [w.sub.e,2] dL]. (30)

On the other hand, the total EWF consumed at the neck propagation stage ([W.sub.e,p]) of the DENT test is only due to neck propagation within the central ligament section of [L.sub.p] in length ([L.sub.p] = [L.sub.0] - [L.sub.i]). That is,

[W.sub.e,p] = 2 [t.sub.0] [[integral].sub.[L.sub.i]/2.sup.[L.sub.0]/2] [w.sub.e,2] dL. (31)

The specific EWF for each stage of crack growth can be determined by dividing Eqs. 30 and 31 by the corresponding ligament area for the crack growth:

[w.sub.e,i] = [W.sub.e,i]/([L.sub.i][t.sub.0]) = (1 + [1/[eta]])[w.sub.e,1] + [w.sub.e,2]

[w.sub.e,p] = [W.sub.e,p]/([L.sub.p][t.sub.0]) = [w.sub.e,2]. (32)

The earlier expressions indicate that [w.sub.e,i] may be much larger than [w.sub.e,p] due to the inclusion of the first term, (1 + [1/[eta]])[w.sub.e,1], provided that difference between [h.sub.1] and [h.sub.2] is not significant.

For HDPE that shows stable necking in the DENT test, further modification is needed for the earlier expressions to take into account the increase of [h.sub.1] and [h.sub.2] with the crack growth. Once [bar.[sigma]] can be expressed explicitly as a function of [bar.[epsilon]], and [h.sub.1] and [h.sub.2] as a function of L, analytical expressions for [w.sub.e,i] and [w.sub.e,p] can then be established to show their dependence on the fundamental material behaviour such as the stress variation as a function of strain. We believe that the analysis will also lead to the development of the resistance curve for the crack growth in the DENT test. This will be studied in the near future.

CONCLUSIONS

Plastic deformation and necking of HDPE under tensile loading were investigated using UT and DENT tests. Results from the UT tests suggest that the neck development in the HDPE can be divided into two stages, for neck inception and neck propagation, respectively, with fracture occurring only at the end of the neck propagation stage. The neck development in the DENT tests was similar, except that the crack growth was involved in both stages of the neck development. By quantifying the energy consumed for each stage of the neck development in the DENT tests, the specific EWF values for each stage of the crack growth were determined using linear regression to zero ligament length. The results suggest that EWF values for the two stages are different, and the EWF values may actually vary continuously as a function of the crack growth length in the DENT test.

The study also observed two interesting phenomena in the DENT tests. One is a constant crack growth speed throughout the whole fracture process, despite two distinctively different stages of the neck development. The other is a fixed fraction of the total ligament length for each stage of the neck development, which is independent of the original ligament length used in the study.

ACKNOWLEDGMENTS 

Kwon also acknowledges the support from Queen Elizabeth II Graduate Scholarship for the study. Sincere gratitude is due to B. Faulkner and A. Yuen who provided invaluable technical assistance for the success of the test program.

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H.J. Kwon, P.-Y.B. Jar

Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8

Correspondence to: H.J. Kwon; e-mail: [e-mail]

Contract grant sponsor: Natural Sciences and Engineering Research Council of Canada (NSERC).