# Combined effect of temperature and thickness on work of fracture parameters of unplasticized PVC film.

INTRODUCTIONThe fracture behavior of brittle materials can be easily characterized using the Linear Elastic Fracture Mechanics (LEFM) approach. For brittle and semi-brittle materials, which exhibit small-scale crack-tip plasticity, an adjustment to the crack length can be made, so that the entire specimen can be assumed to exhibit Hookean elasticity. However, in the presence of a large plastic deformation zone as In toughened and ductile polymers the energy dissipated in this zone can no longer be attributed to fracture process, and this limits the application of LEFM to these range of materials. One method that has been used widely for many years to characterize the fracture behavior of toughened and ductile polymers is J-integral. Another method, which has been used in recent years and has gained popularity owing to its experimental simplicity, is the Essential Work of Fracture (EWF).

According to EWF theory, which was first proposed by Broberg (1) and further developed by Cotterell and Reddel (2), a linear relationship exists between the specific total work of fracture and ligament length, giving a positive intercept at zero ligament length. This positive intercept is termed the specific essential work of fracture, [w.sub.e], which is essentially a surface energy term. Recent studies (e.g. 3-21) have shown that for a given specimen thickness, [w.sub.e] is a material constant being independent of both the size and geometry of the test specimen.

One of the limitations of using polymeric materials at elevated temperatures is that their mechanical properties often change with temperature. The work presented here reports on the combined effects of thickness and temperature on the work of fracture parameters of an uPVC film. This work also considers partitioning of the work of fracture parameters into yielding and necking/tearing components and further examines the reliability of crack opening displacement (COD) as an alternative method of determining [w.sub.e] and its components.

EXPERIMENTAL DETAILS

The material used in this study was an uPVC film supplied in the form of A4 size sheets of nominal thicknesses 0.15, 0.225 and 0.4 mm. The glass transition temperature of this material was approximately 72[degrees]C as measured using differential scanning calorimeter (DSC) technique.

EWF tests were performed on all three thicknesses between 23[degrees]C and 60[degrees]C using Double-Edge Notched Tension (DENT) specimens as shown in Fig. 1. These specimens were prepared by first cutting the A4 size sheets into rectangular coupons of a constant width, W, of 35 mm and length, H, of 70 mm (the longitudinal axis of the coupon was perpendicular to the long edge of the sheet). The coupons were then notched to produce series of double-edge notched tension (DENT) specimens with ligament lengths ranging from 2 to 16 mm. The measurement of the ligament length was carded out prior to testing using a travelling microscope. After notching, each specimen was tested to complete failure in an Instron testing machine at a crosshead displacement rate value of 5 mm/min, using pneumatic clamps with initial separation. Z, of 35 mm (i.e. Z/W ratio of 1). The load-displacement (P-[delta]) curve for each specimen was recorded using a computer data logger.

Measurements of tensile yield stress and modulus were also made on all three thicknesses between 23[degrees]C and 60[degrees]C Tests were carried out on dumbbell-shaped specimens having a constant width of 3.9 mm and a length of 55 mm, within the gauge length region. These specimens were uniaxially loaded in an Instron testing machine at a crosshead displacement rate value of 5 mm/min and produced load-displacement [P-[delta]) curves typical examples of which are shown in Fig. 2. The curves, obtained exhibited a clear yield point (i.e. the maximum load) and drop in load after yield due to strain softening phenomenon. Tensile test results are summarized in Table 1, where it can be seen that yield stress ([[sigma].sub.y]) and the flow stress ([[sigma].sub.flow]) and the Young's modulus (E) all decrease with increasing temperature, merely reflecting the temperature dependence of the viscoelastic response of the material. Results further suggest that values of [[sigma].sub.y] and [[sigma].sub.flow] increase somewhat with increas ing thickness.

RESULTS AND DISCUSSION

Deformation and Fracture Behavior

Figure 3 shows typical examples of EWF load-displacement curves obtained in this study as ligament length and thickness were changed. Curves indicate that, extensive yielding and ductile tearing of the ligament region preceded failure of the DENT specimens tested in the present study. The notable feature of the curves is their geometrical similarity, which is the basic requirement for determining essential work of fracture, i.e., mode of fracture was independent of ligament length. The mode of fracture is believed to be that of plane-stress, as evidenced by the contraction of the specimen surfaces.

DENT load-displacement curves further reveal that value of the total work of fracture (given by the area under the curve) and extension to break increased with increasing temperature, whereas maximum load and the rate at which load after maximum diminished to zero decreased with increasing temperature. These effects would affect the resistance of the material to crack propagation.

Visual observation of the specimens during the test revealed that failure of the specimens occurs after a series of sequential events. These were identified as opening and rounding of the crack tips, formation and propagation of a stress-whitened duckbill-shaped yielded zone at each crack tip, overlapping of the two yielded zones (i.e. full yielding of the ligament region), and ductile tearing of the ligament region. It was noted that the load on the P-[delta] curves reached a maximum when the two plastic deformation zones met midway along the ligament region. At this point, overlapping of the two yielded zones took place, leading to a prominent load drop after maximum, as illustrated by the P-[delta] curves in Fig. 3. Sample separation eventually occurred when the two advancing crack fronts met halfway along the ligament region, forming a diamond - shaped zone at failure.

It is worth pointing out that at 60[degrees]C, the stress-whitened region was not very pronounced, presumably because the test temperature was close to the glass transition temperature of the material, indeed, its was found that by keeping the broken specimens above the glass transition temperature (i.e. 73[degrees]C), the plastic deformation zone diminished and the shape of the specimen was fully restored. This healing process suggested that no substantial change in the initial entanglement network morphology of the polymer structure had taken place because of loading. The recovery of the initial shape further suggested that cold-drawing, and not true plastic flow, took place in the plastic zone of this uPVC material.

Work of Fracture Analysis

According to EWF theory, when failure of the specimen is preceded by extensive yielding and slow crack growth as in this study, a toughness parameter called the Specific Essential Work of Fracture, which may be considered as a material constant for a given thickness, may be evaluated. The EWF approach is based on the assumption that two process zones as shown in Fig. 4a may be identified in a notched specimen. Namely, an inner fracture process zone (IFPZ) where the actual fracture takes place; and an outer plastic deformation zone (OPDZ), where plastic deformation is necessary to accommodate the strain in the inner fracture zone. The work expended in the inner fracture process zone by neck formation and tearing is termed the essential work of fracture, [W.sub.e], and that which is dissipated in the outer plastic deformation zone is termed the "non-essential work of fracture," [W.sub.p]. The total work of fracture, [W.sub.f], is therefore defined as:

[w.sub.f] = [w.sub.e] + [w.sub.p] (1)

The work [W.sub.e] is essentially a surface energy term and is assumed to be proportional to the ligament length area, i.e.

[W.sub.e] = [w.sub.e]BL (2)

where [w.sub.e] is the specific essential total work of fracture.

The work [W.sub.p] is assumed to be proportional to the volume of the plastic deformation zone, [V.sub.p](L, B), i.e.

[W.sub.p] = [w.sub.p][V.sub.p](B, L) (3)

Here [w.sub.p] is the specific non-essential work of fracture.

When plane-stress conditions prevail in the ligament region, it is further assumed that [w.sub.e] is constant and [V.sub.p](L, B) is proportional to [BL.sup.2] with a proportionality constant [beta] that is independent of the ligament length, i.e.

[W.sub.p] = [beta][w.sub.p][V.sub.p][BL.sup.2] (4)

The specific total work of fracture, [W.sub.f] (= [W.sub.f]/LB) is then expressed as:

[w.sub.f] = [w.sub.e] + [beta][w.sub.p] (5)

Equation 5 predicts a linear relationship between the specific total work of fracture, [w.sub.f], and ligament length, L. The positive intercept at L = 0 gives [w.sub.e] and the slope of the line gives [beta][w.sub.p] as shown schematically in Fig. 4b.

It has been suggested (e.g. 4) that for [w.sub.f] to be linearly dependent upon L, the length of the ligament region should lie within a certain range of L values known as the "valid ligament length range." This range has been empirically recommended as:

3B - 5B [less than or equal to] L [less than or equal to] min (W/3, 2[R.sub.p]) (6)

where 2[R.sub.p] is the size of the plastic deformation zone surrounding the crack tip, the value of which may be estimated from the following linear elastic fracture mechanics equation:

2[R.sub.p] = 1/[pi] (E[w.sub.e]/[[sigma].sup.2.sub.y]) (7)

where E and [[sigma].sub.y] are, respectively, the Young's modulus and the tensile yield stress of the material. The minimum ligament length 3B to 5B is proposed to ensure that the state of stress in the ligament region is one of pure plane-stress and not mixed mode, in which case, the stress state would have both plane-stress and plane-strain characteristics. In the mixed mode region, the plastic constraint factor rises with shortening of the ligament length region, and as a consequence of this, [w.sub.f] becomes lower than the predicted value (see Fig. 4b). This transition in the stress state can be verified through a plot of [[sigma].sub.n] versus ligament length, where [[sigma].sub.n] is the net-section stress at maximum load (i.e. maximum load on the P-[delta] curve divided by the ligament area, LB).

Figure 5 shows typical examples of [[sigma].sub.n] versus L plot. It can be seen that although [[sigma].sub.n] decreases as ligament length increases, it is difficult to locate precisely at what ligament length this stress state transition takes place. However, tentatively, it seems that this transition occurs at a ligament length of about 6 mm, below which [[sigma].sub.n] tends to rise with decreasing L. The L/B ratio at which this transition occurs therefore lies between 15 and 48 and not 3 to 5, as recommended. However, when considering the evolution of [w.sub.f] with L as in Fig. 6, we observe no strong evidence of decreasing [w.sub.f] for ligament length values less than 6 mm. In view of this, a question must be raised as to whether the rise in [[sigma].sub.n] with shortening of the ligament region is due to the transitional change in stress-state or is it a reflection of the change in strain-rate in the ligament region? As time to yield decreased with shortening of the ligament length. If the observed behavior is indeed due to stress state transition, then the implications are that either the requirement 3B-5B is not sufficient to guarantee a plane-stress conditions or that increases in constraint factor are small enough not to affect the linearity of [w.sub.f] versus L plots. It is worth noting that since values of the L/B ratio at transition points as reported in the literature (e.g. 6, 11, 13, 15-19), vary from one polymer system to another, the ratio is therefore material dependent and not a universal value, as originally thought.

It is also evident from plots in Fig. 5 that [[sigma].sub.n] decreases with increasing temperature and increases with increasing thickness. According to plasticity theory (22), owing to plastic constraint imposed by the notches, the net-section stress for DENT geometry under plane-stress conditions is expected to rise to 1.15 times the uniaxial yield stress of the material. However, as illustrated in Figs. 6, values of [[sigma].sub.n] at large ligament lengths do not conform to the predicted value of 1.15[[sigma].sub.y]. This observation has also been acknowledged for a wide range of polymeric materials, both of crystalline and amorphous nature (e.g. 11, 13-18, 21).

The prerequisite L < W/3 also played no significant role in this study, as plots of [w.sub.f] versus L remained fairly linear for all values of L tested here. This prerequisite, which is recommended for avoiding the edge effect, has been found by many to be too restrictive. As for prerequisite L [less than or equal to] 2[R.sub.p], which is recommended to ensure that full yielding of the ligament region occurs before crack growth, verification according to Eq 7 can be made only once [w.sub.e] is measured.

Following Eq 5, plots of [w.sub.f] versus L in Fig. 6 were linearly interpolated. Values of [w.sub.e] and [beta][w.sub.p], obtained by extrapolation to zero ligament length and from the slope, respectively, are given in Table 2 for each specimen thickness/temperature. Results indicate that [w.sub.e] is independent of temperature at all three thicknesses, and increases with thickness at all three temperatures. As regards [beta][w.sub.p], it can be seen that this parameter increases with increasing temperature, as expected. However, while between thickness values of 0.15 mm and 0.225 mm, [beta][w.sub.p] increased, it decreased marginally between thickness values of 0.225 mm and 0.40 mm.

Finally, using values of [w.sub.e] given in Table 2, the size of the plastic zone, 2[R.sub.p], was calculated using Eq 7. The calculated values show that 2[R.sub.p] is either close to or exceeds the length W/3.

Partitioning of the Total Work of Fracture

Bearing in mind that full yielding of the ligament region in PVC specimens occurred at maximum load, it was possible as illustrated in Fig. 7, to partition total work of fracture, [W.sub.f], into two components:

(i) Work of fracture for yielding of the ligament region, [W.sub.y]

(ii) Work of fracture for necking and subsequent tearing of the ligament region, [W.sub.nt].

Previous studies (9, 10, 13, 15) have shown that when [w.sub.f] is partitioned in this way, variations of [w.sub.y] and [w.sub.nt] also follow that of Eq 5. Accordingly, by considering the yielding and necking/tearing contributions to [w.sub.f], Eq 5 was then split into the following equations:

[w.sub.y] = [W.sub.y]/LB = [w.sub.e,y] + [[beta].sub.y] [w.sub.p,y]L

[w.sub.nt]= [W.sub.nt]/LB = [w.sub.e,nt] + [[beta].sub.nt] [w.sub.p,nt]L (8)

where [w.sub.e,y] and [w.sub.e,nt] represent, respectively, the yielding and the necking/tearing related parts of the specific essential total work of fracture [w.sub.e]; [[beta].sub.y][w.sub.p,y] and [[beta].sub.nt][w.sub.p,nt] are, respectively, the yielding and the necking/tearing components of the specific non-essential total work of fracture, [beta][w.sub.p], i.e.

[w.sub.e] = [w.sub.e,y] + [w.sub.e,n]

[beta][w.sub.p] = [[beta].sub.y][w.sub.p,y] + [[beta].sub.nt] [w.sub.p,nt] (9)

Figure 8 provides experimental evidence that variation of [w.sub.y] and [w.sub.nt] with L is indeed linear and that the specific essential and non-essential work of fracture are themselves composite terms. It is also evident from these plots that the amount of work contributing to [w.sub.f] by each component depends strongly upon temperature and to a lesser extent upon thickness. It is realized that while [w.sub.y] decreases with increasing temperature, it increases with increasing thickness. On the other hand, [w.sub.nt] increases with both temperature and thickness. In any case, as clearly demonstrated in Fig. 8, [w.sub.nt] is always greater than [w.sub.y], thus implying that a greater proportion of the specific total work of fracture ([w.sub.f]) stems from the necking/tearing part of the fracture process rather than from its yielding part.

To determine values of the specific essential work of fracture terms [w.sub.e,y] and [w.sub.e,nt] and the specific nonessential work of fracture terms and [[beta].sub.y][w.sub.p,y] and [[beta].sub.nt] [w.sub.p,nt], lines describing the variations of [w.sub.y] and [w.sub.nt] with L were extrapolated to zero ligament length according to Eq 8. The intercept values [w.sub.e,y] and [w.sub.e,nt] are summarized in Table 2 and plotted with [w.sub.e] values in Fig. 9 as a function of thickness at two of the temperatures. Plots show that the specific essential work terms increase linearly with increasing thickness at all three temperatures. It can be deduced also that in contrast to [W.sub.e], which shows no significant variation with respect to temperature (for a fixed thickness), values of [w.sub.e,y] and [w.sub.e,nt] both vary significantly with temperature. However, while [w.sub.e,y] decreases, [w.sub.e,nt] increases with increasing temperature. For example, at room temperature, approximately 60% of [w.sub.e] iS c ontributed by the necking/tearing component whereas at 60[degrees]C, this value rises to approximately 90%.

Values of the specific non-essential work of fracture parameters [beta][w.sub.p], [[beta].sub.y][w.sub.p,y] and [[beta].sub.nt][w.sub.p,nt] are also summarised in Table 2 and their dependence on thickness at two of the temperatures is shown graphically in Fig. 10. It can be observed that [[beta].sub.nt][w.sub.p,nt] is always greater than [[beta].sub.y][w.sub.p,y] and that the effect of increasing temperature is to reduce [[beta].sub.y][w.sub.p,y] and to increase [[beta].sub.nt][w.sub.p,nt]. It may be deduced that at 23[degrees]C almost 60% of the total non-essential work is contributed by the necking/tearing component, compared to 90% at 60[degrees]C. Indeed, when additional tests were performed at 70[degrees]C, the outcome was drawing of the ligament region and no crack growth. It can be anticipated, based on the 60[degrees]C, results, that at 70[degrees]C, the contribution made by [[beta].sub.y][w.sub.p,y] was so small that the essential prerequisite for crack growth was not available. Plots in Fig. 10 al so suggest that the specific non-essential work of fracture parameters [[beta].sub.y][w.sub.p,y] and [[beta].sub.nt][w.sub.p,nt] increase with thickness between 0.150 mm and 0.225 mm and decrease between 0.225 and 0.4 mm.

Estimation of the EWF Using Crack Opening Displacement (COD)

Recently, estimation of [w.sub.e] via crack opening displacement has received wider attention by several investigators (11, 13, 15, 20). By and large, the approach has provided a reasonable estimation of [W.sub.e]. Although common to all studies is the observed linearity between the extension at break, [e.sub.b], and ligament length, there remains some degree of uncertainty as to how the information derived from this relationship may be used. The observed linearity between [e.sub.b] and L is expressed as:

[e.sub.b] = [e.sub.o] + [e.sub.p]L (10)

where [e.sub.o], is the intercept value of [e.sub.b] at zero ligament length. This value is taken as a measure of the crack opening displacement Recently Mouzakis et al. (20) portioned [e.sub.b] into yielding ([e.sub.y]) and necking/tearing ([e.sub.nt]) components (see Fig. 7) and in analogy with Eq 10 expressed the variation of [e.sub.y] and [e.sub.nt] with ligament length as:

[e.sub.y] = [e.sub.o,y] + [e.sub.p.y] L

[e.sub.nt] = [e.sub.o,nt] + [e.sub.p,nt] L (11)

where [e.sub.o,y] is the crack opening displacement at the ligament yield point (critical elongation needed to cause crack tip blunting) and [e.sub.o,nt] is the difference between the crack propagation and crack tip blunting stages (i.e. [e.sub.o,nt] = [e.sub.o] - [e.sub.o,y]).

Figure 11 shows typical plots of [e.sub.b], [e.sub.y] and [e.sub.nt] versus ligament length as a function of temperature and thickness, where It can be seen that extension values increase linearly with ligament length. It can also be deduced that [e.sub.b], [e.sub.y] and [e.sub.nt] all increase with increasing temperature and to a lesser extent with increasing thickness. However, the contribution made by [e.sub.nt] to [e.sub.b] is always greater than that made by [e.sub.y].

The extrapolated values at zero ligament length, i.e. [e.sub.o], and [e.sub.o,y] and [e.sub.o,nt] are given in Table 3, where it can be seen that they increase with temperature as well as thickness.

To estimate [w.sub.e] via [e.sub.o] a simple relationship has been used (e.g. 13, 15, 20, 21):

[w.sub.e] = [lambda][e.sub.o][[sigma].sub.y] (12)

Although the above equation has been used with some success by many (e.g. 13, 15, 20, 21) for estimating [w.sub.e], selecting [lambda] has been far from being a straightforward exercise. The following values of [lambda] can be found in literature:

(i) [lambda] = 1

(ii) [lambda] = 1.15 (Hill's plastic constraint factor)

(iii) [lambda] = [[sigma].sub.ns]/[[sigma].sub.y] (experimentally determined plastic constraint factor. [[sigma].sub.ns] = steady state value of the net-section stress derived from the plot of [[sigma].sub.n] versus L)

(iv) [lambda] = 0.67 (parabolic load-displacement curve assumption with no plastic constraint factor)

(v) [lambda] = 0.77 (parabolic load-displacement curve assumption with Hill's plastic constraint factor)

(vi) [lambda] = 0.67[[sigma].sub.ns]/[[sigma].sub.y] (parabolic load-displacement curve assumption with experimentally determined value for plastic constraint factor)

In analogy with Eq 12, the following equations may be written for estimating [w.sub.e,y] and [w.sub.e,nt] via crack opening displacement values [e.sub.o,y] and [e.sub.o,nt]

[w.sub.e,y] = [lambda][e.sub.o,y] [[sigma].sub.y]

[w.sub.e,nt] = [lambda][e.sub.o,nt] [[sigma].sub.y] (13)

Table 3 summarises values of the specific essential work of fracture parameters estimated according to Eqs 12 and 13 with values of [lambda] as given by (i), (iii). (iv) and (vi). It can be seen that by and large, agreement between the directly measured and the estimated values is generally very poor, even though occasionally a reasonable agreement is found for one of the parameters, if not for all. In view of this, we recommend that calculations via COD must be treated with extreme caution, as they could unreasonably be either higher or lower than the directly measured values.

Plastic Zone Shape Factor, [beta]

Measurement for the shape factor can be obtained from the height of the plastic zone (h) versus ligament length (L). For a diamond-shaped plastic zone, we have:

[V.sub.p] (L,B) = [[beta].sub.diamond] [BL.sup.2] = 1/2 hLB (14)

giving a linear relationship between h and L

h = 2[[beta].sub.diamond] L (15)

i.e. the slope of the h versus L plot is equal to 2[[beta].sub.diamond]. Plots of h versus L are given in Fig. 12, where it can be seen that variation is linear over the entire ligament length range, as predicted by Eq 15. The calculated values of [[beta].sub.diamond] are given in Table 2, indicating that [[beta].sub.diamond] is not affected significantly by the thickness of the specimen but increases with increasing temperature. Values of [w.sub.p] calculated as [beta][w.sub.p]/[[beta].sub.diamond] are also given in Table 2, where it can be seen that it decreases with increasing temperature.

CONCLUSIONS

Double edge notched uPVC specimens having thicknesses of 0.150, 0.225 and 0.40 mm were pulled to complete failure at test temperatures of 23[degrees]C, 40[degrees]C and 60[degrees]C, to study the combined effect of temperature and thickness on the fracture toughness. A linear relationship was obtained between the specific total work of fracture, [w.sub.f], and ligament length, L. It was found that the specific essential work of fracture ([w.sub.e]) was independent of temperature but increased linearly with thickness, at all three temperatures. Based on the maximum load on the load-displacement curve, [w.sub.f] was partitioned into the specific work of fracture for yielding ([w.sub.y]) and the specific work of fracture for necking/tearing ([w.sub.nt]). A linear relationship was found for each term as a function of ligament length from which the specific essential work ([w.sub.e,y], [w.sub.e,nt]) and non-essential work ([[beta].sub.y][w.sub.p,y], [[beta].sub.nt][w.sub.p,nt] terms were determined. Results showe d that [w.sub.e,y] and [w.sub.e,nt] also increase linearly with thickness. However, while [w.sub.e,y] decreased, [w.sub.e,nt], increased With creasing temperature, at each thickness. Results further demonstrated that, extension at break ([e.sub.b]) and its yielding ([e.sub.y]) and necking/tearing ([e.sub.nt]) components all increase linearly with ligament length. However, estimations of the specific essential work of fracture parameters ([w.sub.e], [w.sub.e,y] and [w.sub.e,nt]) via the intercept values of [e.sub.b], [e.sub.y] and [e.sub.nt] at zero ligament length were found to be unsatisfactory, as they were either much higher or lower than the directly measured values.

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[Figure 11 omitted]

[Figure 12 omitted]

Table 1 Tensile Test Data. B = 0.15 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [[sigma].sub.y](MPa) 44.1 35.5 19.6 [[sigma].sub.flow](MPa) 34.3 26.8 15.2 E(GPa) 2.93 2.58 1.73 B = 0.225 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [[sigma].sub.y](MPa) 46.6 38.5 21.3 [[sigma].sub.flow](MPa) 35.7 27.4 14.9 E(GPa) 3.12 2.96 2.04 B = 0.40 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [[sigma].sub.y](MPa) 46.4 34.3 21.9 [[sigma].sub.flow](MPa) 37.2 26.1 15.1 E(GPa) 3.07 2.86 2.06 Table 2 EWF Test Results. B = 0.15 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [W.sub.e] (kJ/[m.sup.-2]) 22.06 22.21 21.24 [W.sub.e,y] 8.36 6.16 4.09 (kJ/[m.sup.-2]) [W.sub.e,nt] (kJ/ 13.67 17.28 18.90 [m.sup.-2]) [beta][W.sub.p] 2.05 2.25 3.77 (MJ/[m.sup.-3]) [[beta].sub.y][W.sub.p'y] 0.70 0.49 0.19 (MJ/[m.sup.-3]) [[beta].sub.nt] 1.35 1.65 3.53 [W.sub.p'nt] (MJ/[m.sup.-3]) 2[R.sub.p] (mm) 10.58 14.47 30.45 [[beta].sub.diamond] 0.052 0.062 -- [W.sup.p] (MJ/[m.sup.-3]) 39.42 36.29 -- B = 0.225 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [W.sub.e] (kJ/[m.sup.-2]) 26.51 27.23 26.92 [W.sub.e,y] 10.20 7.20 4.07 (kJ/[m.sup.-2]) [W.sub.e.nt] (kJ/ 15.91 20.43 22.26 [m.sup.-2]) [beta][W.sub.p] 2.53 3.08 4.39 (MJ/[m.sup.-3]) [[beta].sub.y][W.sub.p'y] 1.09 0.68 0.37 (MJ/[m.sup.-3]) [[beta].sub.nt] 1.44 2.36 4.06 [W.sub.p'nt] (MJ/[m.sup.-3]) 2[R.sub.p] (mm) 11.94 17.31 38.53 [[beta].sub.diamond] 0.050 0.061 -- [W.sup.p] (MJ/[m.sup.-3]) 50.60 50.49 -- B = 0.40 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [W.sub.e] (kJ/[m.sup.-2]) 38.50 39.06 39.18 [W.sub.e,y] 14.69 10.73 5.65 (kJ/[m.sup.-2]) [W.sub.e.nt] (kJ/ 24.10 28.03 32.81 [m.sup.-2]) [beta][W.sub.p] 2.12 3.05 4.09 (MJ/[m.sup.-3]) [[beta].sub.y][W.sub.p'y] 0.89 0.87 0.47 (MJ/[m.sup.-3]) [[beta].sub.nt] 1.20 2.45 3.68 [W.sub.p'nt] (MJ/[m.sup.-3]) 2[R.sub.p] (mm) 17.48 30.69 53.57 [[beta].sub.diamond] 0.054 0.063 -- [W.sup.p] (MJ/[m.sup.-3]) 39.26 48.41 -- Table 3 Estimations of the Specific Essential Work of Fracture Parameters Via COD. B = 0.15 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [e.sub.o] (mm) 0.850 0.942 1.07 [e.sub.o,y] (mm) 0.432 0.385 0.352 [e.sub.o,nt] (mm) 0.418 0.557 0.714 [[sigma].sub.y] (MPa) 44.1 35.5 19.6 [[sigma].sub.ns] (MPa) 46 37 21.5 [W.sub.e] (kJ/[M.sup.-2]) 22.06 22.21 21.24 [W.sub.e] (i) 37.49 33.44 20.97 [W.sub.e] (iii) 39.11 34.85 23.01 [W.sub.e] (iv) 24.99 22.29 13.98 [W.sub.e] (vi) 26.07 23.24 15.34 [W.sub.e,y] (kJ/[m.sup.-2]) 8.36 6.16 4.09 [W.sub.e,y] (i) 19.05 13.67 6.90 [W.sub.e,y] (iii) 19.87 14.25 7.57 [W.sub.e,y] (iv) 12.70 9.11 4.60 [W.sub.e,y] (vi) 13.25 9.50 5.05 [W.sub.e,nt] (kJ/[m.sup.-2]) 13.67 17.28 18.90 [W.sub.e,nt] (i) 18.43 19.77 13.99 [W.sub.e,nt] (iii) 19.23 20.61 15.35 [W.sub.e,nt] (iv) 12.29 13.18 9.33 [W.sub.e,nt] (vi) 12.82 13.74 10.23 B = 0.225 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [e.sub.o] (mm) 0.863 1.05 1.08 [e.sub.o,y] (mm) 0.406 0.397 0.260 [e.sub.o,nt] (mm) 0.457 0.652 0.819 [[sigma].sub.y] (MPa) 46.6 38.5 21.3 [[sigma].sub.ns] (MPa) 51 40 26 [W.sub.e] (kJ/[m.sup.-2]) 26.51 27.23 26.92 [W.sub.e] (i) 40.22 40.43 23.00 [W.sub.e] (iii) 44.01 42.00 28.08 [W.sub.e] (iv) 26.81 26.95 15.34 [W.sub.e] (vi) 29.34 28.00 18.72 [W.sub.e,y] (kJ/[m.sup.-2]) 10.20 7.20 4.07 [W.sub.e,y] (i) 18.92 15.29 5.54 [W.sub.e,y] (iii) 20.71 15.88 6.76 [W.sub.e,y] (iv) 12.61 10.19 3.69 [W.sub.e,y] (vi) 13.80 10.59 4.51 [W.sub.e,nt] (kJ/[m.sup.-2]) 15.91 20.43 22.26 [W.sub.e,nt] (i) 21.30 25.10 17.45 [W.sub.e,nt] (iii) 23.31 26.08 21.29 [W.sub.e,nt] (iv) 14.20 16.74 11.63 [W.sub.e,nt] (vi) 15.54 17.39 14.20 B = 0.40 mm Thickness/Parameter 23[degrees]C 40[degrees]C 60[degrees]C [e.sub.o] (mm) 0.943 1.292 1.460 [e.sub.o,y] (mm) 0.419 0.435 0.402 [e.sub.o,nt] (mm) 0.524 0.867 1.06 [[sigma].sub.y] (MPa) 46.4 34.3 21.9 [[sigma].sub.ns] (MPa) 55 47 31 [W.sub.e] (kJ/[m.sup.-2]) 38.50 39.06 39.18 [W.sub.e] (i) 43.76 44.32 31.97 [W.sub.e] (iii) 51.87 60.74 45.26 [W.sub.e] (iv) 29.17 29.54 21.32 [W.sub.e] (vi) 34.58 40.48 30.17 [W.sub.e,y] (kJ/[m.sup.-2]) 14.69 10.73 5.65 [W.sub.e,y] (i) 19.44 14.92 8.80 [W.sub.e,y] (iii) 23.05 20.45 12.46 [W.sub.e,y] (iv) 12.96 9.95 5.87 [W.sub.e,y] (vi) 15.36 13.63 8.31 [W.sub.e,nt] (kJ/[M.sup.-2]) 24.10 28.03 32.81 [W.sub.e,nt] (i) 24.31 29.74 23.21 [W.sub.e,nt] (iii) 28.82 40.75 32.86 [W.sub.e,nt] (iv) 16.21 19.83 15.48 [W.sub.e,nt] (vi) 19.21 27.17 21.91

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Author: | Arkhireyeva, A.; Hashemi, S. |
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Publication: | Polymer Engineering and Science |

Date: | Mar 1, 2002 |

Words: | 6045 |

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