A mechanistic approach for determining plane-stress fracture toughness of polyethylene

Engineering Fracture Mechanics 77 (2010) 2881–2895

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A mechanistic approach for determining plane-stress fracture toughness of polyethylene P.-Y.B. Jar *, R. Adianto, S. Muhammad Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8

a r t i c l e

i n f o

Article history: Received 12 March 2010 Received in revised form 7 July 2010 Accepted 14 July 2010 Available online 17 July 2010 Keywords: Fracture resistance Plane stress Necking Polyethylene DENT

a b s t r a c t A new mechanistic approach is used to characterize resistance of polyethylene to deformation and fracture in double-edge-notched tensile test. The new approach considers all three mechanisms involved in the fracture process, i.e. for fracture surface formation, shear plastic deformation, and necking, and can be used to determine values of specific energy consumption for each mechanism. This is different from the conventional approach, known as essential work of fracture (EWF), which does not consider the difference between shear plastic deformation and necking. Results from the new approach for a polyethylene copolymer show that specific energy density for fracture surface formation is about half of that determined from the EWF approach, and specific energy density for necking is very close to that determined from simple tensile test. The latter provides some support for validity of the new approach in characterizing fracture behaviour of polyethylene when accompanied by large deformation and necking. The paper also points out crack growth conditions that have to be met for valid application of the EWF approach and shows that such conditions are not met when deformation and necking occur in polyethylene. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction For polymers like polyethylene, plane-stress fracture at room temperature is accompanied with extensive ductile deformation that involves a localized neck-forming process. Research for characterizing the fracture resistance is often based on the concept of essential work of fracture (EWF), using data collected from double-edge-notched tensile (DENT) test in the plane-stress condition (e.g. [1–4]). The EWF concept is subject to the deformation scenario that two mechanisms are involved in the testing, one being essential for the fracture surface formation and the other non-essential [5]. Since the corresponding energy consumption is in different orders of specimen dimension (i.e. ligament length), a regression process is used, based on the following equation, to separate the essential part from the non-essential part.

W f =ðlo to Þ ¼ we þ bwp lo

ð1Þ

where Wf is the total work of fracture for the DENT test, lo the original ligament length, to the original ligament thickness, we the specific EWF value that represents the specific energy density for fracture surface formation, wp the specific energy density for plastic deformation, and b the shape factor for the plastic deformation zone. As initially proposed by Cotterell and Reddel [1], the above equation is applicable to fracture processes in which the plastic deformation zone is fully developed before the crack growth commences, and the non-essential part of the energy * Corresponding author. Address: 4-9 Mechanical Engineering Building, University of Alberta, Edmonton, Canada T6G 2G8. Tel.: +1 780 492 6503; fax: +1 780 492 2200. E-mail address: [email protected] (P.-Y.B. Jar). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.07.008

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Nomenclature A fracture surface area bo, bn mesh width in the central ligament region of DENT specimen F, Fmax load measured from the DENT test hn, hn,fract height of the necked region formed in the DENT test parameter, along with bh, for characterizing the neck dimensions in the DENT test ho l, lo, lp ligament length of DENT specimen m unloading stiffness measured from the DENT test, in N/mm t time tn, to ligament thickness of DENT specimens U strain energy Vc crack growth speed at the neck propagation stage Vn, Vneck,p, Vs volume of deformed region in the DENT test cross-head speed for the DENT test Vd Wf, Wf,p total work of fracture measured from the DENT test we specific EWF value for formation of fracture surface, in kJ/m2 wp, wp,n, wp,s specific energy density for the plastic deformation of DENT specimen, in kJ/m3 Greek symbols b shape factor for the plastic deformation zone in the EWF analysis bh parameter, along with ho, for characterizing the neck dimensions in the DENT test d, dc, dfract displacement measured from the DENT test en,DENT necking strain of DENT specimen

2

consumption is a function of lo (note: not just a quadratic function of lo). The assumption has been validated for low-alloy steel plates [1], but for polymers like polyethylene, such validation has rarely been conducted in detail. Although value of bwp from the EWF approach can serve as an indication of the material capacity for plastic deformation, uncertainty of the b value has limited its usefulness. As a result, the EWF approach is mainly used to determine the we value that represents the essential material property for resistance to fracture in a given loading condition (in this case, plane-stress condition). In spite of the lack of validation for its applicability to polymers, the above equation has been adopted by many research groups to determine we values for polymers in ductile fracture [2,3,6–14]. Other studies [8,15,16], after recognizing the complexity of polymer fracture, proposed modifications of Eq. (1) for the measurement of we. However, those modifications are still under the assumption that the non-essential part of energy consumption is a function of ligament-length squared. As to be shown in this paper, this assumption is not applicable to ductile fracture in polymers like polyethylene. Instead of EWF, a new mechanistic approach is presented here to determine fracture resistance of polyethylene in the plane-stress condition. Data from DENT tests of a polyethylene copolymer are used to demonstrate the new approach. This paper will show that in addition to the extensive plastic deformation and necking, shear plastic deformation is also involved in the fracture process, which leads to the formation of a deformation zone with triangular-contoured fracture surface in the central part of the ligament section. Since volume generated by shear plastic deformation is different from that by necking, their energy consumption should not be grouped together as one term for the non-essential part of Eq. (1). Therefore, as to be shown in this paper, Eq. (1) is not suitable for determining the we value when the shear plastic deformation is involved in the fracture process. In the following sections, conditions for mechanical tests conducted in the current study will first be presented, followed by description of the trend shown in the test results. Based on the specimen fracture behaviour, mechanisms involved in the testing will be identified, and then, the new mechanistic approach for determining the we value be given in detail and applied to the same test data set obtained from the polyethylene copolymer. Value of we determined from the new approach will be compared with that from the EWF approach, and validity of the two approaches be discussed from the view point of the assumption required for the EWF approach and the specific necking energy density determined from the new mechanistic approach.

2. Experimental details 2.1. Materials Polyethylene used in this study is a pipe-extrusion grade ethylene–hexene copolymer, provided by NOVA Chemicals. Characteristics of its pellets are listed in Table 1. The polyethylene pellets were compression-molded into plaques of 250  250 mm2 with thickness around 10 mm. The plaques were then machined to DENT specimens of 90 mm wide and

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Table 1 Characteristics of polyethylene copolymer used in the study. Mn represents number-average molecular weight and Mw weight-average molecular weight. Mn

Mw

Branches (1000 C)

Density (g/cm3)

14,400

154,400

4.7–5.3

0.940

210 mm long, with the original ligament length (lo) in the range from 25 to 34 mm at an increment of 3 mm. Three specimens were tested for each ligament length. Prior to the tests, a very sharp tip was introduced to the machined notches using a fresh razor blade, to avoid a blunt notch tip that has been reported to inflate the we value [17,18]. The original ligament length, lo, was measured between sharp tips introduced by the razor blades, using Mitutoyo profile projector with resolution of 0.1 mm. 2.2. Test set-up DENT tests were conducted using a Galbadini Quasar 100 universal testing machine, at a cross-head speed of 5 mm/min. All tests showed a stable fracture process, recorded using D-70 Nikon camera at a rate of 1 photograph per 10 s, for test duration of around 5 min or longer. Change of ligament length in the fracture process was established from those photographs, which was then used to determine the crack growth rate. Note that a rectangular mesh pattern was introduced to the specimens in the ligament area prior to the test, as shown in Fig. 1, to facilitate the use of photographs to quantify the ligament contraction due to the neck development. As to be described in Section 4, the above information was used to determine values of parameters required in the new approach.

Fig. 1. Schematic description of DENT specimens and mesh pattern introduced in the ligament section.

Fig. 2. An example of the loading–unloading tests conducted in the study to determine variation of stiffness during the DENT test.

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Apart from the above specimens that were needed for the ‘‘ordinary” DENT test, five additional specimens, two with lo = 28 mm and three with lo = 34 mm, were tested with intermittent unloading–reloading cycles, to examine variation of unloading stiffness with the crack growth. Each specimen was subjected to five unloading–reloading cycles after the maximum load, two at the neck initiation stage (i.e. with neck growth in the ligament length direction) and three at the neck propagation stage (i.e. with neck growth in the loading direction). A typical load–displacement curve from this type of tests is shown in Fig. 2. Note that the hysteresis loop generated by unloading–reloading was ignored. Instead, the unloading stiffness was determined based on the slope of the dashed lines in Fig. 2. 3. Deformation and fracture behaviour of polyethylene in DENT test Studies (e.g. [3,8,14]) have shown that DENT specimens of many polymers, such as polyethylene, generate extensive necking in the plane-stress fracture, among which Kwon and Jar [8] pointed out that the necking process consists of two stages that are for neck initiation and neck propagation, respectively. As mentioned earlier, the former involves neck growth along the ligament length direction, while the latter in the loading direction. At the neck initiation stage, crack growth is accompanied by the reduction of ligament thickness that progresses gradually from notch tips towards the centre of the ligament, to reach a final thickness tn that is only a small fraction of the original thickness to. This behaviour has also been observed in metallic materials, and is referred to as the transition from plane-strain-like to plane-stress fracture [4,19,20]. Therefore, energy consumption at the neck initiation stage should be excluded from the consideration of plane-stress fracture. At the second stage of the neck development, the neck front advances in the loading direction, with specimen thickness behind the neck front remaining nearly constant. Therefore, crack growth at this stage is through a region of constant thickness, for which the fracture is regarded as in the true plane-stress condition. Although it may be argued that energy consumption at the neck initiation stage might become negligible if sufficiently thin specimens were used for the testing, we believe that the EWF approach is still not suitable for ductile fracture of

Fig. 3. Typical experimental observations: (a) load–displacement curve from the DENT tests, and (b) top, front, and side views of a DENT specimen after the test.

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polymers like polyethylene as its applicability requires assumption that the non-essential part of energy consumption should be a function of ligament-length squared. It will be shown in this paper that this assumption cannot be satisfied. This paper will also show that even by considering the neck propagation stage only, the non-essential part of energy consumption cannot be expressed as a function of ligament-length squared only. Therefore, Eq. (1) is not suitable for the analysis of planestress fracture of polyethylene. To illustrate the deformation and fracture behaviour of polyethylene in the DENT test, a typical load–displacement curve and snap shots of the deformation behaviour for specimens used in this study are presented in Fig. 3. Fig. 3a shows a change of the load-drop rate between the neck initiation and the neck propagation stages. The transition point, point F in Fig. 3a, coincides with transition of the neck development, i.e. from the neck initiation to the neck propagation. The right vertical dashed line in Fig. 3a marks the commencement of the neck propagation at which the neck has been fully developed through the whole ligament section, with the subsequent neck growth only in the loading direction. Note that as depicted in Fig. 3a, load and displacement follow a linear relationship at the neck propagation stage. Only at the very end of the test, prior to the final fracture, does a relatively sharp load drop occur. Fig. 3b shows the top, front and side views of a typical specimen after the DENT test. The top view indicates clearly that the fracture process consists of two stages. At the first stage, i.e. the neck initiation stage, significant thickness reduction was generated from the notch tips towards the ligament centre, but at the second stage, i.e. the neck propagation stage, this did not occur. As transition between the two stages has completed at the right vertical dashed line in Fig. 3a and ligament thickness from this point onwards remains nearly constant, the energy consumption for forming unit neck volume at this stage can be treated as constant. As a result, the non-essential part of energy consumption should be directly proportional to the neck volume formed at this stage. It should be noted that deformation during the neck propagation stage produces a triangular contour in each half of the fractured specimen, which contains a distinct symmetric quadrilateral zone of constant thickness, as depicted in the front view of Fig. 3b. Length of the horizontal diagonal of the quadrilateral zone, lp, represents the total crack growth length developed at the neck propagation stage. Three characteristic behaviours that were observed at the neck propagation stage are summarized below: (1) Crack growth speed The crack growth speed was found to be constant at the neck propagation stage. A typical plot of ligament length versus time is shown in Fig. 4a, in which time is set to be zero at the beginning of the test. The figure indicates that the ligament length decreases with time in an almost linear fashion at each stage of the test, with the rate of ligament length decrease changing slightly during the transition between the two stages. However, the rate of ligament length decrease at the neck initiation stage cannot be used to determine the crack growth speed. This is because the ligament contracts at this stage, thus

Fig. 4. Measurement of crack growth during the DENT test: (a) variation of ligament length as a function of time, with time recording from the beginning of the test, and (b) crack growth speed at the neck propagation stage as a function of the original ligament length (lo).

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contributing to the ligament length decrease. However, such contraction no longer exists at the neck propagation stage, as indicated by the two very right photographs in Fig. 3a. Without the contraction, the change of ligament length can then be used to determine the crack growth speed at this stage. Fig. 4a suggests that the crack growth speed at the neck propagation stage is constant. Only before the final fracture does a relatively fast crack growth occur. The crack growth speed at the neck propagation stage is summarized in Fig. 4b as a function of lo, which suggests that the crack growth speed increases over 10% with the increase of lo from 25 to 34 mm. As change of the crack growth speed over the range of lo used in the study is not negligible, its value needs to be determined experimentally, average of which for specimens of the same lo is then used in the energy balance formulation to determine the specific energy consumption for each mechanism involved in the fracture process, as to be shown in the next section. (2) Load–displacement curve The entire load–displacement curve was found to be scalable. That is, all curves can be converted to a master curve by choosing a proper shifting factor. As demonstrated in Fig. 5, plots of load–displacement curves from specimens of different ligament lengths collapse into a single curve after the load was normalized by the maximum load (Fmax) and the displacement by the displacement at fracture (dfract). As a result, instead of requiring an expression of load versus displacement for each ligament length, all load–displacement curves can be expressed using one function, with Fmax and dfract as the normalization factors that vary with the ligament length. For the polyethylene copolymer used in this study, the function that describes all load–displacement curves at the neck propagation stage is:

F=F max ¼ 0:6296ðd=dfract Þ þ 0:7362

ð2Þ

where F and d are load and displacement, respectively. Note that availability of Eq. (2) is not necessary for the new approach to work. However, its availability has greatly simplified the analysis procedure, as the universal expression avoids the need to determine the load–displacement expression for each ligament length used in the testing. (3) Unloading stiffness The unloading stiffness (m) at the neck propagation stage, determined by the method described in Section 2 using specimens with lo = 28 and 34 mm, was found to be a linear function of the normalized displacement. As shown in Fig. 6, m value

Fig. 5. Load–displacement curves of DENT tests, after normalization with respect to maximum load (Fmax) and displacement at fracture (dfract), respectively.

Fig. 6. Variation of stiffness in the neck propagation stage, plotted as a function of the normalized displacement (d/dfract).

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decreases with the increase of the normalized displacement (d/dfract). The figure suggests that the relationship between m (in N/mm) and d/dfract is independent of the initial ligament length, and can be expressed as:

m ¼ 985ðd=dfract Þ þ 1171

ð3Þ

Again, it should be noted that the above expression is not necessary for the new approach to work. However, its availability has greatly simplified the analysis procedure used in the new approach. 4. The new approach and formulation of energy balance equation The new approach uses energy balance principle to develop an equation that considers all mechanisms involved in the fracture process. Following what has been done in the past, the approach ignores viscoelastic behaviour of polyethylene. Therefore, the unloading stiffness is determined using slope of the dashed lines in Fig. 2. It should be noted that since plane-stress fracture of polyethylene in the DENT test occurred at the neck propagation stage, only mechanisms involved at that stage were considered in the energy formulation. Fig. 7a presents a snap shot of the DENT specimen at the neck propagation stage, showing the significant stretch of the mesh in the loading direction. In the figure, white dashed lines enclose an area where deformation was still actively generated by the loading, which is referred to as the active deformation zone hereafter. Fig. 7b presents a schematic description of mechanisms involved in the active deformation zone, which suggests that in addition to mechanisms (i) and (ii) that are for the fracture surface formation and neck growth, respectively, as identified in the previous work [8], mechanism (iii) (shear plastic deformation) is also involved in the fracture process so that the fracture surface forms a triangular contour, as depicted in Fig. 3b. The corresponding specific energy densities are represented by we, wp,n and wp,s, for forming unit surface area, developing a unit neck volume, and generating a unit shear plastic deformation volume, respectively. Note that although definition of we is identical to that in the EWF approach, Eq. (1), wp,n is different from wp. That is, wp,n represents the specific energy consumption for neck growth in the loading direction, while wp for the formation of plastic deformation zone prior to the crack growth from the notch tips. With constant crack growth speed at the neck propagation stage, rates for neck growth and shear plastic deformation should maintain constant during the test, in order to form the quadrilateral-shaped plastic deformation zone shown in Fig. 3b. As a result, height of the active deformation zone, hn in Fig. 7b, can be expressed as a linear function of time as:

ii

iii i

hn

i

: Fracture surface formation, we

ii : iii

Neck growth, wp,n

: Shear plastic deformation, wp,s

(b) Fig. 7. Depiction of deformation behaviour in the DENT test: (a) a snap shot during the neck propagation stage, and (b) a schematic diagram showing three mechanisms involved in the deformation and fracture process.

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hn ¼ ho þ bh V d t

ð4Þ

where ho and bh are constants that need to be determined based on experimental data, Vd the cross-head speed for the DENT test, and t the time measured from the commencement of the neck propagation stage, i.e. at the right vertical dashed line in Fig. 3a t = 0. It should be noted that ho and bh in Eq. (4) may depend on the ligament length lp, but as to be shown later, for polyethylene copolymer used in this study, ho varies with lp but bh does not. Based on the energy balance principle that energy consumed for the above three mechanisms should be equal to the summation of strain energy change and external work input, we have

we dA þ wp;n dV n þ wp;s dV s ¼ dU þ F dd

ð5Þ

where A is fracture surface area, Vn volume of the fully developed necking zone, Vs volume of the shear plastic deformation zone, U strain energy of the specimen, F load, and d the corresponding displacement. Terms dA, dVn, and dVs in the above equation can be expressed in terms of the crack growth speed at the neck propagation stage (Vc) and the cross-head speed (Vd) as:

dA ¼ 2t n V c dt

ð6aÞ

dV n ¼ tn lbh V d dt

ð6bÞ

dV s ¼ 2tn V c hn dt

ð6cÞ

where tn is the ligament thickness in the fully-necked section and l the remaining ligament length in the specimen at time t. With the assumption of a linear relationship between load and displacement during the unloading, strain energy U stored in the specimen during the test can be expressed in terms of load F and stiffness m as:

U¼

F2 2m

ð7Þ

Therefore,

dU ¼

F F2 dF  dm m 2m2

ð8Þ

By substituting expressions for dA, dVn, dVs, and dU into Eq. (5), we have:

ð2t n V c Þwe dt þ ðt n lbh V d Þwp;n dt þ ð2tn V c hn Þwp;s dt ¼ 

F F2 dF þ dm þ F dd m 2m2

ð9Þ

Since crack growth speed Vc is regarded as constant at the neck propagation stage, time t can be expressed as a linear function of ligament length l as:

t¼

ðlp  lÞ 2V c

ð10Þ

dl 2V c

ð11Þ

Thus,

dt ¼ 

Similarly, displacement d can be expressed as a linear function of l as:

d ¼ dc þ V d

ðlp  lÞ 2V c

ð12Þ

Thus,

dd ¼ 

Vs dl 2V c

ð13Þ

where dc is the displacement at the beginning of the neck propagation stage. After replacing dt and dd in Eq. (9) by the expressions given in Eqs. (11) and (13), respectively, with the removal of the common term dl, the following expression is obtained.

tn we 

1 F dF F2 dm FV d þ  ½wp;n t n lbh V d þ 2wp;s t n hn V c  ¼    2V c m dl 2m2 dl 2V c

ð14Þ

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Through substitution of F, m and hn by the expressions given in Eqs. (2)–(4), respectively, we have:

    1 lp  l Vc wp;n tn lbh V d þ 2wp;s t n ho þ bh V d 2 2V c   V l þ2V d F max 0:3148  d pV c d c c þ 0:7362 þ 0:3148  VVc dd l Vd f f    0:3148  F max  ¼ V d lp þ2V c dc Vd df þ 1170:6 þ 492:5  V c d l 492:5  V c d f f  2 V l þ2V d F 2max 0:3148  d pV c d c c þ 0:7362 þ 0:3148  VVc dd l Vd f f þ  2  492:5  df V d lp þ2V c dc Vd 2  492:5  V c d þ 1170:6 þ 492:5  V c d l f f     V d lp þ 2V c dc Vd Vd þ F max 0:3148  þ 0:7362 þ 0:3148  l   V c df V c df 2

t n V c we 

ð15Þ

The rearrangement by putting together all terms that contain the same order of l yields:

      1 1 1 d 3 2 2 2 a2 gwe þ a2 c þ 2abgwe þ ða2 d þ 2abcÞ l þ b gwe þ ð2abd þ b cÞ l þ l 2 2 2 2 2

2

3

¼ ðaeh  e2 i þ a2 ejÞ þ ½ðaf þ beÞh  2efi þ ða2 f þ 2abeÞjl þ ½bfh  f 2 i þ ð2abf þ b eÞjl þ fjl

ð16Þ

where

a¼

2ilp 2 F max V c

þ mc

ð17aÞ

  b ¼ 2i= F 2max V c

ð17bÞ

c ¼ tn wp;s ð2ho V c þ bh V d lp Þ d ¼ ðwp;n  wp;s Þt n bh V d

ð17cÞ ð17dÞ

hlp Fc þ F 2max V c F max   f ¼ h= F 2max V c

e¼

g ¼ tn V c

ð17eÞ ð17fÞ ð17gÞ

Vd h ¼ 0:3148   dfract F max V d i ¼ 782:2hj ¼ 2 F 2max

ð17hÞ ð17iÞ

and mc and Fc represent stiffness and load, respectively, at the commencement of the neck propagation stage. It should be noted that for all expressions in Eq. (17), apart from parameters c and d that contain unknown values of ho and bh, the other parameters can be determined experimentally. Since l in Eq. (16) is a free variable with values changing from lp to zero at the neck propagation stage, the equation holds only if the following four expressions, each from coefficients of all terms of the same order of l, are satisfied:from lo terms

1 a2 gwe þ a2 c ¼ aeh  e2 i þ a2 ej 2

ð18aÞ

from l1 terms

1 2abgwe þ ða2 d þ 2abcÞ ¼ ðaf þ beÞh  2efi þ ða2 f þ 2abeÞj 2

ð18bÞ

from l2 terms

1 2 2 2 b gwe þ ð2abd þ b cÞ ¼ bfh  f 2 i þ ð2abf þ b eÞj 2

ð18cÞ

from l3 terms

d ¼ fj 2

ð18dÞ

With the expression of d from Eq. (18d), Eqs. (18b) and (18c) can be rewritten as:

2abgwe þ abc ¼ ðaf þ beÞh  2efi þ 2abej

ð19aÞ

2

b c 2 2 ¼ bfh  f 2 i þ b ej b gwe þ 2

ð19bÞ

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After normalization to reduce the coefficient of the first term on the left-hand side of Eqs. (18a), (19a) and (19b) to 1, we have:

1 eh e2 i ej c¼  þ 2g ag a2 g g   1 f e efi ej h c¼ þ þ we þ 2g 2bg 2ag abg g 2 1 fh f i ej þ we þ c¼  2g bg b2 g g we þ

ð20aÞ ð20bÞ ð20cÞ

As shown in Table 2, total values for the terms on the right-hand side of the above three equations are very close to each other. Therefore, those three equations actually represent one relationship between we and c. Consequently, only Eq. (20a) is considered in the following analysis. Through substitution of parameter c by expression of Eq. (17c), Eq. (20a) gives an explicit expression for the relationship between we and wp,s, that is,

we þ

tn ð2ho V c þ bh V d lp Þ eh e2 i ej wp;s ¼  þ 2g ag a2 g g

ð21Þ

Similarly, Eq. (18d) gives an explicit expression for the relationship between wp,n and wp,s:

ðwp;n  wp;s Þtn bh ¼

dF dl

ð22Þ 

2



eh Based on Eq. (21), we and wp,s can be determined by applying linear regression to the plot of ag  ae2 gi þ ejg versus That is, we and wp,s are values of the intersection with the ordinate and the slope of the plot, respectively. The corresponding wp,n value can then be determined from Eq. (22). The remaining barrier for the above approach to work is the determination of ho and bh values in Eq. (4), in order to obtain the explicit expression of hn as a function of time. In this study, bh was determined from the expression below, derived using the same approach as that described in Ref. [21], based on the principle of continuity and incompressibility at the neck growth front.

t n ð2ho V c þbh V d lp Þ . 2g

bh ¼ t o =½to  tn ðbn =bo Þ

ð23Þ

where bo is the original mesh width in the central ligament region and bn the corresponding dimension after the neck is fully developed. With bh value determined, ho can then be calculated from Eq. (4) using height of the quadrilateral region of the post-fractured specimens, denoted as hn,fract/2 in Fig. 3b. By assuming that crack growth speed (Vc) is constant for the entire neck propagation stage, ho can be expressed as:

ho ¼ hn;fract  bh V d

lp 2V c

ð24Þ

Values of bh and ho determined in this way are listed in Table 2, which suggests that with the increase of lo, bh does not change much but ho increases significantly. Note that in the work by Mai and Cotterell [3], ho was assumed to be a function of specimen thickness only. Our results suggest that such an assumption is not applicable to polyethylene when extensive necking occurs. Pardoen et al. [19] suggest that for aluminum ho may vary with the ligament length, but the level of variation in their suggestion is much smaller than that shown in Table 2.It should be pointed out that values of ho and bh determined in this way were based on dimensions of the post-fractured specimens. Therefore, hn in Eq. (4) does not represent the physical Table 2 List of values for right-hand side (RHS) of Eq. (20), bh and ho for all specimens used in the study. lo (mm)

RHS of Eq. (20a)

RHS of Eq. (20b)

RHS of Eq. (20c)

bh

ho (mm)

24.3 24.3 24.3

417 414 409

418 415 409

418 415 409

1.18 1.17 1.18

3.44 3.73 3.76

27.2 27.3 27.3

454 466 459

455 467 460

455 467 460

1.19 1.18 1.17

3.87 4.13 4.39

30.3 30.2 30.2

482 500 482

483 501 483

483 501 483

1.17 1.20 1.19

4.99 4.55 4.77

33.4 33.2 33.2

571 541 513

572 542 514

572 542 514

1.17 1.17 1.17

5.42 5.96 6.31

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Fig. 8. Regression plot of right-hand side of Eq. (21) versus coefficient of the wp,s term in the equation.

Table 3 Results from the new approach, and those from EWF approach based on either total fracture energy or energy partition.

2

we (kJ/m ) wp,n (kJ/m3) wp,s (kJ/m3) bwp (kJ/m3)

New approach

EWF (total work of fracture)

EWF (energy partition)

45 71,200 20,400 –

86 – – 8.7

111 – – 8.1

height of the active deformation zone during the test. Rather, hn is the equivalent height of the active deformation zone after the load is removed. The above approach is under the assumption that Vc remains constant through the entire neck propagation stage. However, experimental data suggest that Vc actually increases slightly before final fracture of the specimen, as indicated by two data points on the right in Fig. 4a. However, since the amount of increase in Vc is small and it occurs only within the last 10% of the crack growth length, it is reasonable to treat Vc as constant throughout the neck propagation stage. Thus, duration of the crack growth at the neck propagation stage can be represented by lp =ð2V c Þ. Despite the assumption being reasonable, further study is needed to examine errors that can possibly be generated by ignoring the change of Vc at the end of the neck propagation stage.     2 t ð2h V c þbh V d lp Þ eh is given in Fig. 8 using values of ho and bh in Table 2, based on which values of  ae2 gi þ ejg versus n o 2g Plot of ag we and wp,s were determined and listed in Table 3, along with wp,n value determined from Eq. (22).

5. Discussion This section will firstly compare we value determined from the new approach with that from the EWF approach. The latter is based on either total work of fracture or energy partition. The comparison will show that those approaches yield very different we values. The comparison will be followed by examining assumptions for the EWF approach, to evaluate its applicability to measurement of polymers like polyethylene. In addition, the wp,n value from the new approach will be compared with that from simple tensile test, to evaluate validity of the new approach for characterizing plane-stress deformation behaviour of polyethylene. In the last part of this section, implication of the new approach for characterizing material fracture resistance and its practical applications will be provided. 5.1. Comparison of we Values Using the same test results as those for Fig. 8, linear regression was applied to the normalized total work of fracture, Wf/ (lo  to), or the normalized work input at the neck propagation stage, Wf,p/(lp  to). The results are presented in Fig. 9 from which we values are determined to be 86 and 111 kJ/m2, respectively, which are also listed in Table 3, for comparison with we value from the new approach. Although those values are very different from each other, it is impossible to assess relevance of those values to fracture toughness of the polyethylene, as no standard method is available to quantify fracture resistance when such large deformation occurs. Nevertheless, the different we values suggest that those approaches yield very different specific energy density for the fracture surface formation.

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Fig. 9. Regression plots based on (a) specific total work of fracture, Wf/(loto) versus lo, and (b) energy partition, Wf,p/(lpto) versus lp.

It should be pointed out that data in Fig. 9 show a better linear correlation than those in Fig. 8. However, good linear correlation itself does not serve as an indication for the correctness of the measurement. Besides, it is not surprising that data in Fig. 8 show larger scattering than in Fig. 9, as the new approach requires determination of parameters such as ho and bh that cannot be measured directly from the experiments. Variation of ho and bh values among specimens of the same ligament length, as shown in Table 2, is believed to be the main source that increases scattering of the data points in Fig. 8. Since it is not possible to evaluate directly the applicability of the above approaches for the fracture resistance of polyethylene, alternative evaluation methods are presented here, first by examining assumption for the EWF approach, and then by comparing the wp,n value (specific energy density for neck growth) obtained from the new approach using DENT test with that from simple tensile test. 5.2. Assumption of the EWF approach As mentioned in Section 1, one of the assumptions for the EWF approach is that the non-essential part of energy consumption for crack growth should be a function of ligament-length squared. That is, the specific non-essential energy consumption (i.e. normalized with ligament area) should be directly proportional to ligament length, with the value being zero at zero ligament length. Unfortunately, for the EWF approach based on the total work of fracture the above assumption cannot be verified directly. This is because deformation is still being developed at the neck initiation stage, therefore, the corresponding specific energy density may vary significantly during the crack growth. In other words, unlike mild steel [1], wp in Eq. (1) does not carry much physical meaning for polyethylene. Furthermore, the significant development of plastic deformation at the neck initiation stage raises serious doubts on whether the total non-essential energy consumption can be expressed as a function of ligament-length squared. Data obtained from the neck propagation stage were used to examine the EWF approach based on energy partition. Since neck has been developed nearly to its full extent at this stage, the specific energy density for the neck formation can be regarded as constant. As a result, the assumption of the non-essential part of energy consumption being proportional to ligament-length squared can be evaluated by checking whether the total necking volume generated at this stage (Vneck,p) is 2 proportional to lp . Based on analysis in the previous section, Vneck,p can be expressed as:

V neck;p ¼ ðhn;fract  ho Þ

lp t n 2

ð25Þ

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With the expression of (hn,fract  ho) given in Eq. (24), we have

V neck;p ¼

bh V d 2 l tn Vc 4 p

ð26Þ 2

Since Vd and tn are independent of lp, bh =V c should be independent of lp if V neck;p is a function of lp . Fig. 10 presents variation of bh/Vc with the change of lp, which clearly shows that the above requirement of bh/Vc being independent of lp is not satisfied. Therefore, the EWF approach based on energy partition is not applicable to polyethylene when extensive necking is involved in the fracture process. 5.3. Specific energy consumption for necking Simple tensile test was used to measure the specific energy consumption for neck growth in polyethylene (named specific necking energy density), from which the value was compared with wp,n determined from the new approach. Although neck growth in the two tests is subjected to different lateral constraint, values of the specific necking energy density at the same strain level should be similar. In this study, necking strain in the DENT test was measured using the dimensional change in the central part of the ligament section. With volume conservation, the true strain required for the neck development in the DENT test (en,DENT) can be expressed as:

en;DENT ¼ ln



bo t o bn t n

 ð27Þ

where bo and to are original width and thickness of the mesh element, respectively, and bn and tn the corresponding values after the neck is fully developed. Fig. 11 summarizes en,DENT values as a function of lp, which suggests that en,DENT can be treated as constant of 1.89 in the range of ligament length used in the study. Specific necking energy density from the simple tensile test was measured using dog-bone specimens (type I of ASTM Standard D638). Tests were conducted at a cross-head speed of 5 mm/min. The final true strain generated in the neck

Fig. 10. Plot of bh/Vc versus lp for examining the assumption for the EWF approach.

Fig. 11. Plot of the necking strain versus ligament length lp.

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Fig. 12. Typical load–displacement curve from the simple tensile test.

was found to be 1.77, about 7% lower than that measured from the DENT test. The corresponding neck growth speed in the simple tensile test was determined using a video clip that recorded the neck development during the test, and was found to be 0.59 mm/min for each neck front. A typical load–displacement curve from the simple tensile test is presented in Fig. 12. The specific necking energy density was determined using the plateau section of the curve. For example, the crosshead displacement of 8.43 mm, equivalent to a testing period of 101.2 s, corresponds to the neck growth distance of 2 mm, i.e. 1 mm for each neck front. With the correction of the cross sectional area due to the plastic deformation before the necking, the specific necking energy density from the simple tensile test was found to be 60,600 kJ/m3, which is about 17% lower than the wp,n value measured from the DENT test (71,200 kJ/m3). However, in view of the smaller fracture strain for the former (1.77 compared to 1.89 in the DENT test) and the significant work hardening expected in the strain range from 1.77 to 1.89, it is reasonable that the specific necking energy density from the simple tensile test is slightly smaller than the wp,n value from the DENT test. If the necking strain in the simple tensile test were increased to 1.89, the corresponding specific necking energy density would have been much closer to the wp,n value. Therefore, the results provide some support to the validity of the new approach for characterizing the plane-stress fracture behaviour of polyethylene. 5.4. Potential applications of the new approach The new mechanistic approach described above provides an alternative to the existing test methods for characterizing material fracture behaviour, especially when large plastic deformation is involved in the fracture process. The study raises at least two issues on the suitability of applying the EWF approach to plane-stress fracture of polyethylene. One is validity of the assumption that the non-essential part of energy consumption is only proportional to ligament-length squared. Results from this study show that this assumption is not valid when necking occurs. Therefore, it is recommended that such an assumption be evaluated whenever necking is involved in the fracture process. The other issue that should be considered is the ignorance of the EWF approach for the deformation development during the crack growth. As mentioned earlier, Eq. (1) was developed for fracture processes that have plastic deformation zone being fully developed before the crack growth commences. However, most materials do not fracture in such a way. Instead, plastic deformation usually continues to develop during the crack growth. In such cases, validity of Eq. (1) is questioned even if regression analysis shows good linear correlation among the data points. The new mechanistic approach described here depicts a general methodology that allows the consideration of deformation development during the crack growth. Using DENT test of polyethylene, the study demonstrates significance of the difference in the we value determined from the new approach from that using the EWF approach. However, instead of applying blindly an equation for characterizing the fracture resistance, such as Eq. (21), it is recommended that mechanisms involved in the fracture process and the corresponding energy consumption be considered case-by-case, in order to formulate the proper energy balance equation that takes into account all mechanisms involved in the fracture process. We believe that in addition to the characterization of fracture resistance, the major advantage of the new mechanistic approach over any traditional method is that the former can be used to quantify resistance to each type of deformation, which has not been achieved before. Such information is extremely useful for establishing criteria needed for computer simulation of deformation and fracture behaviour when complex loading conditions are applied.

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6. Conclusions A mechanistic approach has been developed to characterize fracture behaviour of polyethylene in the plane-stress condition. Observation during the test suggests that only at the second stage of the test, i.e. when the neck growth is in the loading direction, can the deformation be considered as in the plane-stress condition. Three mechanisms (fracture surface formation, neck growth, and shear plastic deformation) were observed in the fracture process. By applying linear regression to experimental data based on the energy balance principle, specific energy consumption for each of the three mechanisms was quantified. Fracture resistance, in terms of we value, determined from the new approach was found to be very different from that using EWF approach based on either total work of fracture or energy partition. Test results suggest that assumptions for the EWF approach are invalid for the DENT test of polyethylene, especially when neck development is involved in the fracture process. Specific necking energy density, determined from simple tensile test, was found to have a value close to wp,n from the new approach. This provides some support to the validity of the new approach for characterizing the plane-stress fracture behaviour of polyethylene. Further study is being conducted to examine applicability of the new approach to other polymers that involve large deformation and necking in the fracture process. Acknowledgements The work was sponsored by Natural Sciences and Engineering Research Council of Canada (NSERC) and NOVA Chemicals, through Collaborative Research and Development (CRD) Program and Discovery Program. Muhammad also acknowledges the support from Queen Elizabeth II scholarship for part of the study. Polyethylene plaques used in the study were provided by G. Yim in NOVA Chemicals, from which test specimens were prepared by D. Waege and D. Pape at the University of Alberta. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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