International Journal of Fracture 69:371-377, 1995. 371 © 1995 Kluwer Academic Publishers. Printed in the Netherlands.

The fracture mechanics of slow crack growth in polyethylene

N. BROWN and X. LU Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6272, USA

Received 13 April 1994; accepted in revised form 8 November 1994

Abstract. The following theoretical equation was obtained for the rate of initiation 6 of slow crack growth in polyethylene:

- trvK4(1 - 72) 2

~ldE2tr2 '

where tr = applied stress, K = stress intensity, 7 = Poisson's ratio, E = Young's modulus, trc = stress to produce a craze, try = yield point, d = primordial thickness of the craze and r/= the intrinsic viscosity of the fibrils of the craze. The dependence of ~ on K agrees with the experimental data. The experimental values of~ vary by a factor of 10 7 depending on the type of polyethylene. This large variation in 6 is directly related to intrinsic viscosity ,/ which evolved from the theory.

1. In troduct ion

Chan and Williams [1] observed that the rate of slow crack growth (SCG) in high density polyethylene can be described by

it = A K 4, (1)

where A is a constant for a given resin and temperature, and K is the stress intensity [2]. More precise data by. Lu and Brown [2, 3] showed that for high density polyethylene, the initiation rate for SCG,

/~ = A / ( 4"7. (2)

Lu and Brown [4] showed that h and ~ have the same dependence on K . Considering the overall scatter in the data from both investigations, both (1) and (2) are in agreement with respect to the dependence on K . Chan and Williams [1] attempted to explain the reason for the exponent 4. Their explanation was based on a model that involved the spacing of defects in the resin and the diffusion of these defects. This explanation was never seriously pursued because different polyethylene resins contain different concentrations of pigments and particle sizes and yet they all agree with (1) and (2). Craze growth theories by Williams and Marshall [5] and Kinlock [6] which are based on relaxation controlled growth, do not apply because their exponent in (1) would vary from 10 to 40 when based on the relaxation times for the elastic modulus and the yield point in polyethylene.

372 N. Brown and X. Lu

, - , 1 5 0 0 El

© I 0 0 0

5 0 0

I I

42°C 5 MPe tf = 527,000"

1 I

Y

O I 0 10

I t~ I

2 0 ~ 0

T I M E ( 1 0 4 m i n ) Fig. 1. CTOD versus time.

I I 4 0 5 0

I ot f

In this paper a fundamental theory is presented for the rate of slow crack growth which agrees with the precise experimental data represented by (2). The theory also gives all the parameters that determine the constant A.

2. Theory

The theory is based on a combination of experimental observation and on fracture mechanics. The basic phenomenon of SCG in PE is shown in Fig. 1 where the crack tip opening dis- placement [CTOD] 6 is plotted against time. A craze forms at the bottom of the notch while the specimen is loaded. The thickness of the base of the craze, is the CTOD which equals 60 shortly after the specimen is loaded, and 6B when fracture initiates at time, tB. Thus, the rate of SCG 6 is given by

_ 6B - - 6 0 (3) tB

As shown in [2, 3] generally (6B -- 60) varies slightly with stress intensity so that either tB or may be used as the measure of SCG. ty, the time for complete failure, is proportional to tB

for a given resin. Thus, tB, t i , or 1/6 all have the same dependence on K in (2). Extensive observations by Lu and Brown [7] show that fracture is initiated in the fibrils at the base of the craze. The model in Fig. 2 illustrates the basis for the theory, el is the stress on the fibrils at the base of the craze. Since SCG is caused by the slow flow of a viscous material, it is assumed that the Newtonian flow law applies:

g _ cTf (4) 60 r/ '

Slow crack growth in polyethylene 373

o"

O"

I l l o;

Fig. 2. M o d e l for SCG. tr = appl ied s t ress , trB = s t ress on b o u n d a r y o f the craze, a! = s t ress on fibrils.

where g is the strain rate of the fibrils and r] is the viscosity of the fibrils. The stress on the boundary of the craze aB is related to af by

t~ 0 a f = as-d- , (5)

where d is the primordial thickness from which craze originates, d/6o is equal to the cross- section area of the fibrils per unit area of craze boundary.

When the specimen is first loaded, aB is obviously about equal to the yield point. Wang, Brown and Fager [8, 9] measured the stress along the boundary of crazes and found that aB was nearly constant along the boundary and approximately equal to the yield point (Fig. 3).

These results suggest that

t~ 0 a s = au--~-. (6)

The stress af can be larger than that of a u because the fibrils are strengthened by being oriented.

Combining (4), (5), and (6)

_ 52a~

n d . (7)

374 N. Brown and X. Lu

<

©

INOTCH TI~ 1 5 r

10

5

I D CRAZE TIP

A B C

T R 4 1 8

A COD = 86/..zm B COD = 94./.sm C COD =116 /.zrn D COD =138 /u..m

I

D ̧

A

I I I 0 1 0 0 2 0 0 5 0 0

P O S I T I O N ( i f . m )

Fig. 3. Stress distribution along boundary of a craze.

Based on the Dugdale theory and experimental measurements of the CTOD by Bassani, Lu and Brown [10] and for plane strain conditions (Fig. 4)

~5 0 - K2( 1 - 3 ' 2 ) E a c ' (8)

where 7 is Poisson's ratio, E is Young's modulus and crc is the stress to produce the craze. Combining (7) and (8)

~} = auK4(1 - 7 2 ) 2 (9)

rldEZcr 2

The dependence of ~ on a and K from the theory closely agrees with the experimental results observed by Lu et al. [2, 3, 10] in (2).

Generally the viscosity is related to temperature by

?7 = riO eQ/RT, (10)

also

K = Y a a 1/2, (11)

241 1 i I

Slow crack growth in polyethylene

I I I

375

=t.

©

<

2 0

16

12

8

4

&

0

0

o

S o A TENSION ~o=115 Ko 2 o BENDING 80= 68 K~

0 ..

42°C

I,~ I I I I I I

0 0 . 0 4 0.08 0.12 0.16 0 .20 0 . 2 4

• (STRESS I N T E N S I T Y ) 2 K(~ ( M P a m I / 2 ) 2

Fig. 4. CTOD versus K 2 for notched bend specimen and notched tensile specimen under plane strain conditions.

where Y is a function of (a/w). Combining (9), (10), and (11)

[0.y(1 - 72)2Y 4] ---- e - 07E- 2c J 0.4 a2 e-O/RT" (12)

The following general results have been reported by Brown et al. for many varieties of linear polyethylenes - both homopolymers and copolymers:

= C ~ n a m e - Q / R T (13)

where n = 3 to 5, m = 1.2-2. Q ~ 100KJ/mol for all polyethylene resins. The factor C is what distinguishes one resin from the other since it can vary by a factor of 107 and is very sensitive to the molecular and morphological structure of the resin. The factor in the square bracket in (12) shows all the parameters that determine the factor C. The only term that can account for the wide variation in C is 7/0. The other material parameters 7, d, E, ac, 0.u and Q do not vary appreciably from one resin to another compared to the fact that C can vary by a factor of 107. ~/0 will be called the intrinsic viscosity of the fibrils.

It is interesting to calculate specific values of r/0 for a copolymer and for a homopolymer using (7) and experimental data. d is the only quantity in (7) which cannot be measured directly. However, the porosity of the craze P is given by

d P = 1 - - - (14)

~50"

376 N. Brown and X. Lu

P can be estimated from microscopic observations. In a typical experiment with a homopoly- mer tested at 30°C and for a yield point of 20MPa, tS0 -- 7 # m and 6 -- 2 x 10-3 #m/s. Microscopic examinations indicate the porosity is about 0.5 so that do = 3.5 #m. Substituting these values in (7) gives r /= 1.4 x 1011Pa s. In a typical experiment with a copolymer at 80°C for a yield point of 4MPa, 60 = 345 #m, ~ = 4 x 10 -3 #m/s . Again, assuming the porosity is 0.5, d = 172#m. Then ~7 = 1012pas . It is now interesting to obtain the intrinsic viscosities, ~/0, by means of (10) and letting Q = 100 KJ/mol.

~10 (homopolymer) = 1.4 x 1011 e-100'000/Rx303 = 8 X 10-7pas ,

~0 (copolymer) = 10lEe -100'000/Rx353 - 1.6 x 10 -3 Pas.

Thus the intrinsic viscosity of the copolymer is 2 x l03 greater than that of the homopolymer. It now remains for a theory based on molecular mechanics and the details of the molecular

and morphological structure to explain the values of the intrinsic viscosity rlo which have been obtained from the present theory.

3. Summary

The resistance to SCG of a resin can be measured by any one of three quantities

1. ~, the rate of CTOD prior to fracture initiation or 2. tB, the time for fracture initiation, or 3. the time for complete fracture.

These three measures are related to each other and have the same dependence on stress, stress intensity and temperature. ~ is the most fundamental measurement because it is directly related to the basic mechanism of fracture.

~ primarily depends on two quantities:

1. 60, the initial CTOD which is related to the extension of the fibrils of the craze at the base of the notch where fracture initiates and

2. ~/0 the intrinsic viscosity of the fibrils at the base of the craze.

is proportional to 62 which in turn is proportional to K 4. The long time weakening of

the fibrils obeys Newtonian flow were ~/60 = af/~l. The long time stress on the fibrils ay is proportional to the applied stress after the stress concentration of the notch is removed by blunting and by the relaxation of the stress field around the notch. The final theoretical equation for ~ is

[ G y ( 1 - - f 2 ) 2 y 4" 34a2

which agrees with the experimental results for homopolymers of polyethylene where

: CGna m e - Q / R T

where 2 . 6 < n < 5 a n d l . 3 < m < 2 . Since C can vary by a factor of 10 7 depending on the molecular and morphological structure

of the resin, this large variation is associated with r/0, the intrinsic viscosity of the fibrils in the craze.

Slow crack growth in polyethylene 377

Acknowledgments

The research was sponsored by the Gas Research Institute. The Central Facilities of the M R L

as supported by the National Science Foundation under Grant No. DMR 91-20668 were most

helpful.

References

1. M.K. Chan andJ.G. Williams, Polymer24(1983) 234. 2. X. Lu and N. Brown, Journal of Materials Science 21 (1986) 2217. 3. Ibid, 4081. 4. Ibid, 2423. 5. J.G. Williams and G.P. Marshal, Proceedings Royal Society of London A342 (1975) 55. 6. A.J. Kinlock, Materials Science 14 (1980) 305. 7. X. Lu and N. Brown, Journal of Materials Science 25 (1990) 29. 8. X. Wang, N. Brown and L. Fager, Polymer 30 (1989) 453. 9. Ibid, 1457.

10. J.L. Bassani, N. Brown and X. Lu, International Journal of Fracture 38 (1988) 43.