Ductile crack growth in poly(ethylene terephthalate) film

Ductile crack growth & poly(ethylene terephthalate) film P. [. VINCENT

A study of crack growth in a poly(ethylene terephthalate) film has been made to check the applicability of fracture mechanics and to devise a suitable technique for assessing the fracture toughness of ductile thermoplastics. A combination of mechanical tests and microscopic examination has shown that fracture mechanics, in a simple single-parameter form, cannot be applied to this material. It is concluded that, because careful tensile tests on unnotched specimens can give so much useful information, it is worth considering whether tests on sharply notched specimens are desirable.

INTRODUCTION

SINCE THE pioneering work by Berry I in 1961 some of the fundamental studies of fracture in thermoplastics have followed the Griffith-Orowan-Irwin approach known as fracture mechanics 2. Sharply notched or cracked specimens have been used, stress distributions and fracture toughness being calculated by assuming linear elasticity. For the more brittle polymers, such as polystyrene and poly(methyl methacrylate), this approach has unified the behaviour of sharply notched specimens with different shapes 3 and has demonstrated the overwhelming contribution to fracture toughness of high but localized strains1, 4. However, the majority of commercial thermoplastics are less brittle than these two materials; in standard tensile tests at r o o m temperature it is common to observe load-extension relations which are far from linear. It appears unlikely that the fracture behaviour of such ductile materials can be treated satisfactorily by theories which assume linear elasticity. Nevertheless, because of the increasing popularity of this approach, it seemed desirable to make a detailed examination of crack propagation in a ductile thermoplastic in order to discover more about the uses and limitations of linear elastic fracture mechanics for this important class of materials. This paper describes the results. Some of the work summarized here has been published in more detail in two reports with limited circulation 5, 6 which are available on application to the author.

TEST MATERIAL

The material selected for this work was a commercial biaxially drawn and thermally crystallized poly(ethylene terephthalate) film about 25 tzm thick. This material is both mechanically and optically anisotropic. It has three different principal refractive indices 7, fl and ~ where 7 > / 3 > a. The particular sample used was specially selected so that/3 was in the extrusion direction, 7 was in the transverse direction and a was perpendicular to the film plane. 534

DUCTILE CRACK GROWTH IN PET FILM Figure 1 shows the non-linear tensile stress/strain curve obtained when a parallel-sided strip was extended at r o o m temperature; the specimen was 1 c m wide, 10 cm between the clamps and was stretched in the :v direction at 5 m m per minute. N o crazes, deformation bands or necks were observed in the specimen during extension; this is thought to be a consequence of the fact that there is no drop in load before fracture. 300

200

g-Z t/) t¢l

tO

IOO

O

~ 0

I0

20

30

40

SO

Stroin (°/o)

Figure 1 Tensile stress/strain curve for the test material in the 7 direction with the recovery curve from 19 ~ strain. The elongation to break is between 57 ~ and 62 NOTCHING TECHNIQUE The actual point on the load/extension curve at which fracture occurs, when a parallel-sided strip is stretched, depends to some extent on accidental defects in the specimen which are difficult to control or quantify. In the fracture mechanics a p p r o a c h a sharp notch or crack is inserted in the specimen; this controlled defect is so severe that it overrides the accidental defects, and its length can be measured. To obtain reliable results it is necessary to ensure that the tip o f the notch is clean and sharp and that the material adjacent to the notch has not been permanently deformed. F o r the 25/zm thick film used in this study parallel-sided strips were notched with a new razor blade while the specimen was held under a tensile strain o f about 1 ~o. The tip radius o f a notch p r o d u c e d in this way was found to be too small to measure in an optical microscope and there was no evidence o f residual deformation. 535

P. I. VINCENT

Equally good results were not obtained when notching other materials in the same way. In a thicker film of the same material the crack front tilted sideways as it advanced ahead of the razor blade. In films from several other polymers it was difficult to avoid permanent deformation in front of the notch tip.

CRACK GROWTH PHENOMENA AND MACROSCOPIC STRESSES When a notched specimen of the test material is extended, the crack grows in a controllable manner in the sense that the growth can be halted by removing the applied deformation. A previous report 5 has shown that the results of tests on notched strips extended in the/3 direction cannot be accounted for by theories based on average stresses or on the linear elastic assumption. Table I Table 1 Tests on single-edge-notched strips extended in the 7 direction Initial notch depth (mm)

Net stress (MN m -z)

Stress intensity factor (MN m -8/z)

Crack growth (tzm)

4-42 4.34 4.40 4.63

60 56-6 50 45

9.40 8.71 7.72 7.35

147 75 31 9.5

1-67 1.81

85

6.71

63

1.78 1.41 1.38 1.87

75 65 61.2 55 50

6.16 5.33 4.43 3.93 4.16

32 21 14 7-6 4.2

0.36 0.39 0.37

79.2 71.1 60.4

2.89 2.70 2.24

13.2 6.9 5.2

summarizes some further tests on single-edge-notched strips extended in the 7 direction. The specimens were extended at constant speed until the required load was reached; the specimen was then unloaded and the crack growth was measured with a microscope. Figure 2 shows the amount of crack growth as a function of the net stress on the reduced section. The results fall into three groups depending on the initial depth of the notch. It is obvious that these measurements cannot be unified by considering only the net stress; there is, however, an indication that the amount of crack growth is unmeasurably small when the net stress is less than 40 M N / m 2 over the whole range of notch depths used. Figure 3 shows the amount of crack growth as a function of the stress intensity factor calculated using the correction factors given by Paris and Sih 7. As before 5 it is clear that these results cannot be accounted for by considering the stress intensity factor calculated by linear elastic fracture mechanics. Furthermore, a comparison of Figures 2 and 3 suggests that the 536

DUCTILE CRACK GROWTH IN PET FILM net stress is a more useful parameter than the stress intensity factor; the curves o f Figure 3 do not suggest that the initiation of observable crack growth occurs at a single value of the stress intensity factor. Rice 8 has stated that linear elastic stress intensity factors are useful when 'the yielded zone at the tip is small c o m p a r e d to characteristic geometric dimensions'. He goes on to say that such 'small-scale yielding solutions have been f o u n d to be highly accurate approximations to available complete

150 140 130 120 I10 I00 9O -.1

t"-

o

8O 7O 6O 50

2

u

40 30 20 I0 0 0

I

I

I

I0

20

30

I

40

50 6 0

70

80

90 I 0 0

Net. stress (MN rn-2)

Figure 2 Crack growth as a function of average net stress on the reduced section for various initial notch depths: O, 4'344'63 rnm; x, 1'38-1.87 mrn; D, 0.364).39 mm solutions up to substantial fractions (typically, one-half) o f general yielding loads'. F r o m Figure 1 the stress/strain relation becomes grossly non-linear at a stress a r o u n d 100 M N / m 2 which may be taken as the yield stress in the ~, direction. F r o m Table I and Figure 2 it can be concluded that the small-scale yielding solution is not an adequately accurate approximation when the net stress is over 40 ~ o f the yield stress. Evidently there is only observable crack growth when the average stress is so large, relative to the yield stress, that the 537

P.

I.

VINCENT

yielded zone is large and the linear elastic stress intensity factor is no longer a useful measure of the stress field. It is possible to obtain some appreciation of the size of the yielded zone by stretching a notched specimen and observing it between crossed polars. Figure 4 is a micrograph of such a specimen taken by monochromatic green light. At this stage the crack had grown forward about 30/zm. By the methods to be discussed in the next section, it can be shown that the outer isochromatic fringe corresponds to a local strain of 5 ~ and is therefore inside the yielded 150

O !

140

I I

I

130

I I I I I I I I I I

120 I10 I00 90 A

i=

::L ,.,C

o CD ~J

,o

80

I

P

70

I I I !

60

I t

50

!

40

g

30

!

20

I I I

j

I0 0 0

×

6

I

1

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9

Io

Stress intensity factor (MN in -3/2)

Figure 3 Crack growth as a function of stress intensity factor: initial notch depths-©, 4.34-4.63 mm; ×, 1.38-1-87 ram; IS],0-36--0.39 mm zone. It is about 17/zm ahead of the crack tip and this figure gives a measure, though an underestimate, of the size of the yielded zone. Figure 5 is a micrograph of the same specimen taken after the crack growth was about 150/zm; at this stage the centre of the isochromatic fringe is about 98/~m ahead of the crack tip. As the crack grows, the yielded zone expands. Because the dominant contribution to the fracture toughness comes from the strain energy in the yielded zone, the expansion of this zone implies that the fracture toughness increases as the crack grows. This is also evident from 538

DUCTILE CRACK G R O W T H IN PET FILM

Figure 4 Isochromatic fringes for a crack growth of 30 e.m ( x 153)

iiiiiljI¸ iilt!iiii,'ili~ii~il

Figure 5 lsochromatic fringes for a crack growth of 150 t,m ( • 153) 539

P. I. VINCENT

macroscopic tests 5 and from Figures 2 and 3. The fracture toughness of a sharply notched specimen of a ductile thermoplastic cannot be characterized by a single number. Rice 8 has explained that such: 'growth effects occur because of the advance of a crack into plastically deformed material. The effect is most easily seen in the fully plastic deformation of a rigid-plastic (or nearly so) material under imposed boundary displacements. This may be contrasted with a non-linear elastic material having similar uniaxial monotonic tension behaviour, for which the advance of a crack would cause the strain field to re-adjust so that a large concentration remains at the tip'.

Figure 1 shows the recovery curve obtained when a parallel-sided strip was stretched by 19 ~ before the machine was reversed at the same speed. The material is neither rigid-plastic nor non-linear elastic but, from the large size of the hysteresis loop, it seems sufficiently plastic to account for the occurrence of growth effects. Close examination of the notch tip in Figure 5 shows that the tip has tilted sideways during growth. This suggests the possibility that the crack is growing in the tearing mode (anti-plane strain) rather than in the opening mode (plane strain or plane stress)7, 9. (The author is indebted to Professor K. E. Puttick for pointing out this possibility.) With due care in the preparation of the original notch, no crack tilting is observed before the crack growth reaches about 60 tzm, before which all the other effects described in this paper may be measured. There is, therefore, no evidence that the conclusions of this work are affected by the presence of additional shear stresses near the notch tip. STRAIN

ANALYSIS

Because the amount of crack growth is not determined by either the average net stress or the linear elastic stress intensity factor, it is necessary to find some technique for determining the actual stress or strain distribution during crack growth. It was shown previously a that this can be achieved by photoanalysis. Tests made on unnotched specimens under strain showed that the change in optical path difference was proportional to the axial strain and was independent of the time under load and whether the measurements were made during extension or recovery. Figure 6 shows the relation between axial strain in the y direction and the optical path difference, which is proportional to the birefringence (?-/3). Comparing Figures 1 and 6 it is obvious that the relation between optical path difference and stress is nonqinear and multivalued. This finding suggests that changes in optical path difference in notched specimens under stress can be used to measure strain distributions. First, however, it is necessary to consider two points of detail. (I) When the direction of the strain at any point is neither the local ~, direction nor the local/3 direction, there is a tendency for the ~, direction to rotate towards the applied strain. This rotation complicates the strain analysis unduly and it is therefore necessary to use this photo-analysis of 540

DUCTILE C R A C K G R O W T H IN PET FILM

strain only where it is clear from symmetry that the local strain is always along either the 7 or/3 direction. Measurements were therefore made only on specimens cut in the 7 or/3 direction, notched at right angles to the axis of the specimen and only at points on the line of symmetry between the notch tip and the opposite edge of the specimen. (2) The calibration is obtained on a specimen subjected to a uniaxial tensile stress whereas it is known t° that in a tension test on a notched specimen of an isotropic elastic material the state of stress near the notch tip is biaxial (or even triaxial for a thick specimen). An experiment was performed to check the significance of this possible complication. If the extension ratios are 1800 I

1600

E 1400 U c

1200

t'-

o o.

I000

800, 700

0

I

I

1

I

I

I

I

I

2

4

6

8

I0

12

14

16

I

18 2 0

S t r a i n (Olo)

Figure 6 A calibration curve of optical path difference against axial strain in the 7 direction denoted by '~1 in the extension direction, A2 in the film plane, perpendicular to the extension direction, and A3 perpendicular to the film plane; then A], Ae and the change in optical path difference were measured directly on an unnotched specimen stretched in the y direction. A notched specimen was also stretched in the ~, direction and held at constant extension. The change in optical path difference was measured as a function o f x (the distance from the notch tip). Then the transmission interference pattern was used to measure A3 as a function of x . Z~l as a function of x could then be estimated in two different ways : (a) Assuming that the relation between A] and change in optical path difference is the same in the unnotched and notched specimens. 541

P. I. V I N C E N T

(b) Assuming that the change in optical path difference is proportional to (A1 -- Az) and that A1A2Aa =- 1.

Figure 7 is a graph of hi deduced by method (a) against A1 deduced by method (b) at the same point in the notched specimen. It can be seen that there is no evidence of systematic deviation between the two methods. 1.25

A~

1,20

x

._o 1.15

r"

•~. 1.10 o 5 1.05

1"00

i

I'00

1.05

I

1.10

l

1.15

1.20

1'25

X1(not allowing for biaxiality)

Figure 7 A graph showing that it is not necessary to allow for biaxiality of strain (see text for details) Accordingly, it seems reasonable to use only the far simpler method (a). Of course this is not generally true in photo-elasticity but there is a special reason why it is permissible in this case: the material is anisotropic and the thickness change under stress is not more than a few per cent. Thus the strain state at a point in a notched specimen is similar to that in an unnotched specimen though the stress state may be different. I f a notched specimen is held under strain and viewed between crossed polars through a microscope, it is possible to measure the optical path difference as a function of distance from the crack tip to within about 1/~m of the tip. Using the calibration graph (Figure 6) this can be converted into a graph of axial strain (~) against distance from the crack tip (x) along the line of symmetry. For a material which obeys the linear elastic theory, a graph of 1/E2 against x should be a straight line passing through the origin. It was shown in reference 5 that, for notched specimens extended in the/3 direction, the graph of 1/e 2 against x was approximately straight but that, when extrapolated, it did not pass through the origin; E was found to have a finite value, denoted by Emax, at x = 0. Also, because of the expansion of the yielded zone, the strain at a given x increases as the crack grows. Figure 8 illustrates this 542

D U C T I L E C R A C K G R O W T H IN PET FILM

24

20 16 o

12

t-

P u~

_

\X'~x ~ X~~ x

x-----_~

8

4 0

I

0 5

I

I

0

I

15

I

I

I

20 25 30 35

Distance from crack tip(l~m)

Figure 8 Curves showing the distributions of strain near the crack tip for specimens stretched in the 7 direction. Two different amounts of crack growth: O, 10t~m; x, 97~m

Table 2 Changes in strain distribution for four specimens stretched in the ~, direction Specimen width (mm) Notch depth (mm) Crack growth (/~m)

20 9.21 102

Distance from crack tip (/~m) 1.3 2.6 3.9 5.2 6.5 13.0 18.0 26.0 33.0 39.0

20 0-47 100

1.07 0'10 101

10 2.36 97

Strain (%) 25.4 17.9 15.8 14.3 13.4 10.4 9.1 6.9 6.2 5.8

26.1 20.2 18.3 15.2 14.3 11.3 9.7 8.4 6.5 5.0

26.2 18-1 17.3 15-1 14.7 11.4 9.9 8.6 8.4 7.3

24.2 19.7 17.2 14.6 13.4 10.6 9.1 7.4 7.0 5.8

p o i n t for two specimens stretched in the 7 direction. It was also stated in reference 5 that the strain d i s t r i b u t i o n was n o t affected by certain changes in initial n o t c h a n d specimen dimensions. Table 2 illustrates this p o i n t for four specimens stretched in the 7 direction. The specimen width was varied by a factor o f 20 a n d the notch depth was varied by a factor of 90 b u t the a m o u n t of crack growth was held c o n s t a n t at a b o u t 0.10 mm. Differences between the t a b u l a t e d strains can be seen to be relatively insignificant. The basic principle of fracture mechanics is that fracture occurs at a given 543

P. I. V I N C E N T

stress (or strain) distribution near the crack tip; this critical distribution can be represented by a single parameter which is independent of specimen dimensions. A similar principle appears applicable to this ductile thermoplastic but in a more complex form. One has to say that a certain amount of crack growth is related to a given strain distribution which is independent of dimensions but which cannot be represented by only one parameter.

FRACTURE CRITERION In reference 5 it was suggested that emax, being independent of specimen and notch dimensions, and of crack growth above 60/zm, was a possible fracture criterion. The fact that Emax was less than the breaking strain of unnotched specimens was attributed to the different stress state. It now seems preferable to consider the fracture criterion suggested by McClintock and Irwin 9 that the strain reaches a critical value at a distance ps from the crack tip. ps is regarded as a structural size or the limit at which 'one can no longer regard the material as a homogeneous plastic continuum'. The work of Yeh and Gei111 suggests that the value of ps for drawn poly(ethylene terephthalate) is of the order of I0 - s m . If this is so, then the extrapolation suggested in reference 5, though appearing small, in fact covers two orders of magnitude in x and so the values of Emax are unlikely to be accurate. This point is made clear in Figure 9 where strain is plotted against x (on a log scale) for two

60 50 40

\\ .c

30

P

U3

20 I0

I

0 i0-e

\\ \~x'-,,x \ ", o \X~xx a'°~ ~ ~,,x~ \o o i ],,o o"

i0-7

lO=S

i0-s

Distance from crack tip (m) (log scale ) Figure 9 Strain as a function of distance from the crack tip on a log scale

different amounts of crack growth. Straight lines are drawn from the point 57 ~ (taken as the break point of unnotched specimens) and 10 -8 m (taken as ps) through the points representing the experimental results. It can be seen 544

DUCTILE CRACK GROWTH IN PET FILM

that these results are not adequate to verify the suggested criterion. It would be necessary to develop a technique for measuring strains at between 10 ~ and 10 8 m from the crack tip.

DISCUSSION

The original object of this work was to find a satisfactory technique for assessing the resistance to fracture of a ductile thermoplastic film which could be used both for the prediction of service performance and for the relation of mechanical properties to micro-structure. Tests on sharply notched specimens were selected for study on the grounds that they would provide reproducible results, that any real material contains defects at which fractures originate and that satisfactory theories were available for deducing useful parameters from the results of such tests. In fact, the main product of the work has been a better understanding of the limitations of this type of approach. In order to obtain reproducible results it is necessary to spend a good deal of time on careful notching and refined measurements. Although it is true that real materials contain defects, they are quite different from artificial notches in size, shape and resultant stress field; consequently, the results of tests on sharply notched specimens cannot readily be used for predicting service performance or for understanding the effect of changes in microstructure. Although a satisfactorytheory is available for brittle fracture, when the net stress is below half the yield stress and the yielded zone is negligibly small, the theoretical problem is much less tractable when it becomes necessary to consider higher stresses, larger yield zones, non-linear stress/strain relations, partial and time-dependent recovery and anisotropy. An essential part of the analysis of the stress field near a notch or crack in a ductile material is a knowledge of the details of the stress/strain relations obtained on unnotched specimens. At the very least, therefore, it is necessary to measure tensile load/extension curves for test materials. Careful tensile tests can also provide the elongation to break which is a measure of fracture in the presence of real, rather than artificial, defects. Much useful information about service performance and the effects of micro-structural variables can be obtained from careful tensile tests and it is worth considering whether tests on sharply notched specimens, with their many additional complications, are really justifiable.

CONCLUSIONS

(1) Measurements of crack growth in a ductile thermoplastic cannot be unified by considering average stresses or by linear elastic fracture mechanics. (2) The size of the yielded zone and the fracture toughness are not constant but increase as the crack grows. This seems to be a consequence of hysteresis. (3) The actual strain distribution can be measured to within about 1 t~m of the crack tip by a photo-analytical technique. It differs significantly from the distribution assumed in linear elastic fracture mechanics. 545

P. I. V I N C E N T

(4) The m e a s u r e d strains are i n d e p e n d e n t o f specimen a n d notch dimensions, within the range examined. (5) It w o u l d be necessary to m e a s u r e strains m u c h closer to the crack tip in o r d e r to discover the true fracture criterion. (6) So m u c h useful i n f o r m a t i o n can be o b t a i n e d f r o m careful tensile tests t h a t it is d o u b t f u l w h e t h e r tests on s h a r p l y n o t c h e d specimens are o f m u c h value.

ACKNOWLEDGEMENT The e x p e r i m e n t a l w o r k in this final stage was carried out b y M r P. White.

Imperial Chemical Industries Limited, Plastics Division, Welwyn Garden City, Hertfordshire, UK

(Received 10 March 1971)

REFERENCES 1 Berry, J. P. J. Polym. Sci. 1961, 50, 107 2 Sih, G. C. and Liebowitz, H. 'Fracture', (Ed. H. Liebowitz), Academic Press, New York and London, 1968, Vol II, pp 67-190 3 Marshall, G. P., Culver, L. E. and Williams, J. G. Plastics and Polymers 1969, 37, 75 4 Berry, J. P. J. Polym. Sci. (A) 1965, 3, 2027 5 Vincent, P. I., Picknell, S. and Harding, G. F., Technical report 73 from Division of Polymer Science, Case Western Reserve University, Cleveland, Ohio, 1967 6 Vincent, P. I., Technical Report 97 from Division of Polymer Science, Case Western University, Cleveland, Ohio, 1968 7 Paris, P. C. and Sih, G. C. 'Fracture toughness testing and its applications', ASTM Special Technical Publication No. 381, 1965, p 44 8 Rice, J. R. 'Fracture', (Ed. H. Liebowitz), Academic Press, New York and London, 1968, Vol II, pp 191-311 9 McClintock, F. A. and Irwin, G. R. 'Fracture toughness testing and its applications', ASTM Special TechnicalPublication No. 381, 1965, p 95 10 Coker, E. G. and Filon, L. N. G. 'A treatise on photo-elasticity', Cambridge University Press, 1957 11 Yeh, G. S. Y. and Geil, P. H. Macromol. Sci. 1967, B1,251

546