a theory for environmental craze yielding of polymers at low temperatures

JOURNAL OF POLYMER SCIENCE: Polymer Physics Edition VOL. 11, 2099-2111 (1973)

A Theory for Environmental Craze Yielding of Polymers at Low Temperatures

NORMAN BROWN , Department of Metallurgy and Materials Science and Laboratory for Research on the Structure of Matter, University of

Pennsylvania, Philadelphia, Pennsylvania i9i 74

synopsis It has been recently discovered that polymers craze at low temperatures in the pres-

ence of nitrogen or argon. A quantitative theory has been developed which explains (1) the critical temperature above which the phenomenon disappears, (2) the critical stress for nucleating a craze, (3) the effect of strain rate on the yield point and size of crazes, (4) the drop in the load during craze yielding, and (5) the increase in strength of the polymer in Np or Ar at high strain rates so that the ultimate strength may exceed that in He or vacuum. The crazing action of the gases is described qualitatively at the molecular level.

INTRODUCTION

It was discovered by Parrish and Brown1P2 that polymers craze at low temperatures around 80°K when nitrogen or argon is present. In helium or under vacuum at 80°K no crazing was observed. Olf and Pe terha found that Hz, O,, and CO, caused isotactic polypropylene to craze. There are several quantitative theories4-' to explain craze yielding, but none of them explains the large dependence of low-temperature environmental craze yielding on the strain rate. Brown and co-workers have found that the stress-strain behavior of all polymers, crystalline and amorphous, is affected by Nz and Ar at low temperatures as compared to He or vacuum environments. 0

In this paper the general experimental features of low temperature environmental crazing will be summarized first. Then theory will be presented to explain the outstanding experimental observations.

SUMMARY OF THE EXPERIMENTAL OBSERVATIONS

1. Nz and Ar exert an environmental effect on the tensile s t r e s t r a i n curves of all polymers at low temperatures.

2. NZ and Ar usually cause craze yielding, whereas in He or in V ~ C U O the polymers undergo brittle fracture or yield without crazing.

3. The effect of N, and Ar does not occur above 150-200"K. 4. The Nz and Ar act at or very close to a free surface.

2099

@ 1973 by John Wiley & Sons, Inc.

2100 N. BROWN

Fig. 1.

Fig.

0 5 10 STRAIN (%I

Stress-strain curves of polycarbonate at about 80°K under vacuum.

in Na,

a n

I 20

0.002/ MIN.

c .- H * - 15 - in in W

k in

a

0

and

1 , . . . I . . , I 10

STRAIN, %

2. Stress-strain curves of polycarbonate at 77°K in NZ at various strain rates.

5. The tensile strength incremes with strain rate in NP or Ar but not in He or vacuum.

6. At low strain rates the tensile strength in N2 and Ar is usually less than in He or under vacuum, but at very high strain rates the polymer tends to be stronger in N2 or Ar.

ENVIRONMENTAL CRAZE YIELDING 2101

Figure 1 shows the typical difference between stress-strain curves in Nz or Ar and He or vacuum at 80°K. In He and under vacuum no crazing is observed. In Nz and Ar, profuse, deep crazes occur as described in great detail by Rabinowitz and Beardmore! Brown and co-workers have found

STRAIN, Y

Fig. 3. Stress-strain curve of polycarbonate at 78°K in He at various strain rates.

COMCARI8ON OC WELD BLMAVIOR I N LIQUID NITROOLN AN0 MLLIUY I

-3- c s , k & l ' ;r' ' io ' iz ' & Z i ' 7.6 '

Fig. 4. Dependence of tensile strength on strain rate (0) in Nz and (0) in He for several polymers at 77°K.

2102 N. BROWN

that the effect of N2 and Ar does not occur above 200°K. More precise measurements by Baers have shown that the effect in N2 does not occur above about 150°K. That the effect rapidly occurs or disappears with the rapid addition or removal of N2 was demonstrated by Parrish and Brown.' This last result indicates that the N2 and Ar act at or very close to a free surface of the polymer. Figure 2 shows the large strain-rate effect in N2 which is also typical for an Ar environment, and Figure 3 shows no strain- rate effect occurs in He or under vacuum. The strain-rate dependence of the tensile strength shown for several polymers in Figure 4 indicates that at high strain rates the tensile strength in N2 tends to exceed that in He. Very high strain-rate data on PMMA by Rabinowite and Beardmores show that the fracture stress in N1 at a strain rate of 103/min is 32 ksi compared to about 20 ksi in He. Recent work by Fischer in our laboratory has shown that nylon at low strain rates of about 10-2/min has a higher tensile strength in N2 than in He, but as the strain rate is reduced, the tensile strength in N2 approaches that in He. Thus it is seen that the tensile

* strength in N2 may be greater or less than in He, depending on the strain rate.

THEORY

Adsorption of N2 and Ar on Polymers The adsorption isotherms of gases like N2 and Ar on polymers have been

determined by Graharn'OJl and Braught, Bruning, and Sch01e.'~ At about 80'K the surface free energy of polymers was decreased by the physical adsorption of N2 and Ar by about 15-40 erg/cm2. The surface free energy of all polymers is in the neighborhood of 50 erg/cm2 and is determined by the van der Waals and hydrogen bonding between the molecules. Thus, it is seen that the adsorption of N2 and Ar at about 80°K significantly lowers the surface free energy of polymers by about 2575%. There are no data on the adsorption of He on polymers at about 80"K, presumably because the effect is negligible. The smaller helium atom has a lower polarieability than N2 and Ar and should not adsorb as readily.

The concentration of adsorbed gas or the coverage can be estimated from the following adsorption isotherm where, for less than a monolayer coverage,

C, = C, exp { -AH/kT] (1) C, is the planar concentration in the gas or the coverage when AH = 0 and AH is the change in enthalpy for adsorption. Since the gas is treated as ideal, we can write

(2) C. = (Pd/kT) exp { - A.H/lcT)

where P is the gas pressure and d is the approximate diameter of a gas molecule. Thus, as is expected, thercoverage decreases with decreasing pressure and increasing temperature, and correspondingly the environ-


mental effect follows the coverage. If it is assumed that the environmental effect becomes unmeasurable when the coverage is less than one-half monolayer, then the pressuretemperature condition at which the environmental effect becomes negligible can be calculated from eq. (3).

1/2 = (Pd3/kT)exp(-AH/kT) (3) The experimental data1°-12 show that AH is not very dependent on coverage and is not very different for N2 and Ar. In the following calculations an average value of - 1650 cal/mole was used for AH as obtained from measurements of the adsorpiion On taking the diameter of a nitrogen molecule as 3.8 A, the environmental effects of N2 and Ar have been calculated to disappear above a temperature of about 157 OK when the pressure is one atmosphere. Obviously, the calculation depends on the degree of coverage where the environmental effect is negligible. Since the change in surface energy is approximately proportional to the coverage when it is less than a monolayer, one-half monolayer coverage roughly corresponds to a change of about 12-38% of the surface energy of the polymer. For another example, 1/10 of a monolayer corresponds to about a 3-7% change in surface energy and should occur at a temperature of 207°K at 1 atm. Our experimental results and those of Olf and Peterlin3 and of Baers show that the environmental effect of Nz and Ar disappears above 150-200°K. Larger molecules, like C02, which have a greater enthalpy of adsorption are expected to be effective at a higher temperature, as verified by our experi- ments and those of Olf and Peterlin.3

The Stress to Nucleate a Craze All of the theories and experimental observations agree that crazing is

nucleated by a cavitation process, one of dilatational yielding. The hydrostatic tension component of the stress is most important in causing crazing, as Sternsteinla pointed out. Using electron microscopy, Kambour and Holick,14 Behan et al.,l5 and Murray and Hulll6 showed that cavitation precedes the main craze. It is likely that the stress concentration a t the apex of the sharp craze promotes the dilatational stress that produces the cavities in front of the craze. Many investigation^'^-^^ have indicated that the crazes generally start at surface flaws. The question to be answered is why polymers craze in N2 and Ar but not in He or under vacuum at low temperatures.

As suggested by Andrews and Bevan15 the stress to nucleate a cavity has two parts: (1) the stress to increase the surface area and (2) the stress to deform the polymer plastically. With a small modification of the equation of Andrews and Bevanl5 the stress to nucleate a cavity is given by

(4)

where uc is the applied stress to produce a cavity; q,, is a stress concentration factor; y is the surface energy of the polymer; B8 is the factor which reduces the surface energy by adsorption; r is the radius of the cavity; uy is the

q&c/3) = (278s/r) + Ca,Bp

2104 N. BROWN

yield point for shear yielding; p,, the factor which reduces uy, is governed by a possible plasticizing effect of N2 and Ar; and C is a constant equal to about 4 or 5. It is assumed that since N2 and Ar reduce the surface energy, they may also act as plasticizers. The condition for shear yielding is given by

qsu. = VYP* (5) where qs is a possible stress concentration factor for shear yielding and q. is about unity, as shown by Parrish and Brown, who found that notched and unnotched specimens of polycarbonate yielded at the same stress. Also, 8, in eq. (5) is essentially unity, because the N2 and Ar only penetrate the polymer to a very small depth, so that bulk shear yielding will not be affected by the presence of the Nz or Ar. If there had been a gross solvation of N2 or Ar in the polymer it would have been reflected by a change in Young’s modulus. The experimental results showed that Young’s modulus WBS the same in N2 and Ar as in He and vacuum. Thus eq. (5) reduces to

us = u y (6) The ratio of the stress to nucleate a craze to the shear yield point is given by eq. (7):

(7) where

UC/.Y = (3/qJ[(2YBB/T%) + C8,l

C = 2/3(1 + ln[E/3(1 - v)uY])

We assume that T = 2 A, y = 50 erg/cm2, and uy = 1.8 X log dynes/cm2 (Fig. 1). As shown by Brown,22 the ratio of Young’s modulus to uy, E/uy , is about 30 to 60 for all polymers. Measurements of Poisson’s ratio in our laboratory as a function of strain and temperature indicate that an average value of v of about 0.4 is most likely. Consequently, C has a value of 4.1 to 4.5. Therefore, eq. (7) becomes

0

uC/uy = (3/qC)(2.8O8 + 4.30,) (8) In the absence of N2 or Ar, 8. = 8, = 1 and uc/uy = 21/qc. Thus, if qc is less than 21, uy is less than uo, and the polymer yields in shear instead of by crazing. Since N2 and Ar reduce the surface energy of polymers by 25- 75%, then 1/4 5 pa < a/1 if sufficient N2 or Ar gets into the polymer a t the point where the craze nucleates. There are no experimental data for evalu- ating @,, the plasticizing effect of N2 or Ar. Letting 0, vary from 0 to 1, in the Nz or Ar environment, we have

(9)

In summary, craze nucleation is favored over bulk-shear yielding, the greater the stress concentration at the surface and the greater the reduction in surface energy and yield stress by the N2 or Ar. The N2 or Ar can affect craze nucleation by penetrating the polymer to a very small distance below the surface, whereas they can only affect bulk shear yielding by deeply permeating the bulk of the specimen.

W q c 2 Q C l U Y > 2/qc


The Mechanism of Craze Yielding After the stress becomes high enough to nucleate crazing in accordance

with eq. (4), the crazes grow, causing craze deformation and the subsequent drop in load at the yield point. It is the purpose of this section to qualitatively and quantitatively describe the mechanism of craze yielding in order to show the basic parameters which determine the yield point and the dependence of the yield point on the strain rate.

As a craze grows, it increases in length and depth, but essentially main- tains a constant thickness except for a taper at its edge, according to the detailed studies of craze m o r p h o l ~ g y . ~ J * J ~ * ~ ~ Rabinowitz and Beardmores showed that the crazes at low temperatures have the form of semielliptical platelets of constant thickness, except a t the edge. As the craze grows, its thickness remains essentially constant, being about 0.1-0.5 p according to the electron micrographs by Kambour and Holick14 and Behan et d.15 The fully developed craze consists of fibrillated polymer with a density of about 50%. Since the plane of the craze is perpendicular to the tensile axis and the displacement produced by the craze is parallel to it, the connection between craze strain and craze growth is given by eq. (10)

de,/dt = N b h / d t

where e, is the craze strain, N is the density of crazes, b is the displacement normal to the craze (and is related to the craze thickness), and a is the area of the craze. It is assumed that the shape of the tapered edge of the craze remains constant during growth, or its change in shape is negligible compared to the overall change in area of the craze. As a first approximation, it is assumed that the shape of the craze remains constant during its growth. Thus, we have

2a = f p 2 (11)

where f is a shape factor and p characterizes the size of the craze. ing eqs. (10) and ( l l ) , we obtain

Combin-

de,/dt = N b f p dp/d t

where dp/d t is now the velocity of the craze front. Equation (12) is similar to that proposed by Hoare and Hull.4 The subsequent theoretical de- velopment is not significantly affected if d p / d t is viewed as an average velocity along the craze front.

The experimental observation^'^^ show that the presence of Nz or Ar is necessary for the maintenance of craze growth. Therefore it has been concluded that the rate of diffusion of N, to the edge of the craze determines the velocity of craze growth. Thus,

dp/d t = AD/J (13)

where A is the adsorption coefficient which varies from zero to unity, D is the diffusion coefficient, and J is the jump distance for an Nz molecule. The coefficient A is nearly zero a t temperatures above that calculated in

2106 N. BROWN

Fig. 5. Illustration of a craze based on electron microscopic observations.~~-~~

0-

Fig. 6. Model for tip of craze showing wedging action of N* under stress u.

eq. (3) above. It is important to consider the diffusion coefficient D and the jump distance J in some detail, because they are not simple quantities. As shown in Figure 5, the morphology in the tapered edge of the craze varies continuously from the fully fibrillated structure containing 100-200 8 voids to the morphology of the bulk polymer. Therefore D and J must vary accordingly. Region I11 of the craze in Figure 5 is modeled in Figure 6 a t the atomic level in order to describe the diffusion of N2 into the tip. There is a region W of the tip outside which the voids and the connecting region between voids are of dimensions equal or greater than the diameter of the N2

molecule (3.8 8). Nitrogen diffuses extremely rapidly into this exterior region. At about 80°K and 1 atm., the surface of the exterior void region is saturated with N2, and in order for N2 to Muse into the region W, a very much slower diffusion process, which approaches that of bulk diffusion, begins. When the N2 molecule jumps a distance J into a cavity smaller than itself, the cavity expands under the stress. It is this very expansion of the cavity which causes the craze to grow. The craze grows when the intermolecular van der Waals bonds rupture. The N2 molecules act as a wedge. A more detailed analysis of the rupturing of the intermolecular bonds indi-


cates that the region W where the Nz causes the crage to grow in conjunction with the applied stress is very narrow. Therefore the values of D and J which are applicable to eq. (13) correspond to the region W where the wedging action of the N, takes place. In the region of the tip, interior to W, the diffusion coefficient is so small that the N, concentration is practically zero. In PMMA, for example, at 80°K the diffusion coefficient of N, in the bulk polymer is about ~ m ~ / s e c . ~ ~ Thus, as illustrated by Figure 6, outside of the wedging region W, the voids are saturated with Nz, and within it there is hardly any. In W the diffusion coefficient D is much greater than that in the bulk, and J is somewhat greater. Values of D and J will be estimated below.

The effect of the stress during craze growth will now be considered at the molecular level as illustrated by Figure 6. Equation (4) is a relation based on a continuum viewpoint. In eq. (4), qc applies to the stress concentration a t the original surface of the specimen. The introduction of the factors ps and ppl which reduce the surface energy and the yield stress, was done on a formal basis. It is not clear that the wedging action of the Nz, as visualized at the molecular level, permits a separation of the surface energy effect from a plasticizing effect. If one visualizes the voids as being “hairy” a t the molecular level as proposed by RosenjZ5 then the distinction becomes even less clear. The experiment~’.~ show that after crazes are formed, the stress by itself cannot keep the craze growing. Both Nz and stress are re- quired for craze growth. As illustrated by Figure 6, the stress tends to open the craze and thereby increases the diffusion rate of Nz. When Nz molecules jump into a small cavity, they wedge it open in cooperation with the stress. Since the Nz is also strongly adsorbed at the surface, it also exerts a pressure to wedge open the craze in the region W. If, in addition, Nz permits the polymer molecules to slide more readily past each other, then one might say the Nz is also a plasticizing agent.

If the stress enhances the diffusion of Nz, the dependence of D on stress is given by

D = Do expf- [Q - ( a v / 3 ) ] / k T ) (14)

where Do is the pre-exponential factor, Q is the activation energy for Nz diffusion, and v is the activation volume for diffusion. The stress is divided by 3 because the hydrostatic component of the stress tensor is deemed most important in this instance. Combining eqs. (12), (13), and (14) gives

de,/dt = (NbfpA Do/J) exp f - Q/kT) exp { ru/3kT] (15)

Equation (15) describes the main features of the craze yielding as exhibited by the stress-strain curve.

For stresses less than the critical stress [eq. (4)] to nucleate the first craze, the deformation is entirely elastic with the elastic strain rate equaling the total strain rate imposed by the testing machine. When the stress exceeds the critical stress, the craze strain rate as described by eq. (15) enters. The

The stress-strain curve is obtained at constant total strain rate.

2108 N. BROWN

TABLE I Estimated Parameters from Figure 4

Polymer B X lP, cme/dyne v, As

PC PMMA Oriented PET (8 = 90')

0.99 1.19 2.74

323 388 893

stress continues to rise as long as the elastic strain increases. As the stress increases, more crazes are nucleated in accordance with the density and distribution of surface flaws. As crazes grow, p increases. Meanwhile the increasing stress increases the velocity of the crazes until the craze strain rate matches the total strain rate. At this point the elastic strain rate becomes zero and the stress reaches a maximum at the yield point. Then N becomes constant, but p continues to increase as the crazes grow. There- fore, the stress drops because the craze strain may equal but not exceed the overall strain rate. The above description .fits the stress-strain curves shown in Figure 2 and corresponds to the description of craze yielding presented by Hoare and Hull.' In many ways the general description is the same as that given by Johnston and Gilman= for the dynamic yielding of crystalline solids by a dislocation mechanism.

The effect of the strain rate on the craze yield point can also be under- stood in terms of eq. (15). At the yield point the craze strain rate is equal to i , the strain rate imposed by the machine. Also the craze strain is given by

e, = Nbfp2/2 (16)

(17)

(18)

(19)

Therefore at the yield point, we have

2 = (2s,AD0/pJ) exp { - Q/kT) exp {uou/3kt)

In i = l n ( 2 s ~ ~ ~ exp(-~/kT)/pJ) + ( v / s ~ T ) ~ , or

The experimental data (Fig. 4) have the form

l n i = M + Buo

where M and B are constants. The forms of eqs. (18) and (19) agree if the parameters in eq. (18) are independent of i or u, and consequently

v = 3kTB (20)

(21) In(2eADo exp { - Q/kT) / p J ) = M

Since Figure 2 shows that ec varies with 1, it can only be stated that there is approximate agreement between the theoretical expression [eq. (18) 1 and the experimental observations. The variation of the size of the crazes with i at the yield point is not known. It is expected that the other parameters in eq. (17) are independent of i. It is of interest to estimate the parameters in eq. (18) by using eqs. (20) and (21) and the data in Figure 4.


Theories for the diffusion of gases in polymers all indicate that the activation volume is much larger than the volume of the diffusing gas molecule. The volume of an N2 molecule is about 54 Aa from adsorption data. The ratio of the activation volumes in Table I to the volume of an N2 molecule varies from 6 to 16. Kuminsn has shown that the theoretical predictions of this ratio range from about 4 to 22, depending on the size of the gas molecule. The theoretical value of this ratio for Nz is estimated (by using Kumin~’~’ calculations) to be about 10, in near agreement with the results of Table I.

The magnitude of the diffusion coefficient, Do exp( -Q/kT) , can be estimated from eq. (21). At a strain rate of i = 2.0/min-l, ec is 4% (Fig. 2). Measurements of crazes8 indicate that p = 0.02 cm. Adsorption data show A * 1 at 80”K, and therefore (Do/J) exp(-Q/kT) = 0.4 X 10-lo cm/sec. The jump distanceJ can be estimated from the activation volume, where the product of J and the cross section of the Nz molecule equals tk activation volume. If the cross section of an Nz molecule is about 14 A2, J is 23 A. Therefore, the diffusion coefficient which controls the rate of crazing is about lo-’’ cm2/ sec. By comparison, the diffusion coefficient in the bulk polymer is cm2/sec at 80°K.

To summarize: Crazing starts a t a critical stress [eq. (4) 1, but the stress increases until the yield point which corresponds to the condition where the rate of crazing matches the head speed of the testing machine. The rate of crazing is controlled by the diffusion of N2 into the polymer. Since the diffusion rate varies exponentially with the applied stress, the yield point is linearly related to the logarithm of the strain rate. The adsorption of the Nz on the polymer not only increases the rate of permeation, but it also decreases the surface energy, so that the Nz molecules wedge open the polymer in cooperation with the applied stress.

From Figure 4, for PC, M is found to be - 18.5.

The Blunting Effect of N2 or Ar At high strain rates the tensile strength in Nz or Ar tends to be greater

than in He or under vacuum (Fig. 4). At these high strain rates, fine crazes are observed. Thus, crazes are nucleated, but the fracture stress is greater with the small crazes than it is in the complete absence of crazing. In the absence of crazing, the fracture stress is governed by original surface flaws. When crazing occurs a t these flaws, the flaw is blunted as indicated previ- ously3 (Fig. 7). If the craze is shallow, then the fracture stress is increased by blunting. As the craze grows, the fracture stress decreases until it falls below the value corresponding to the original surface flaws. Rabinowitz and Beardmore* have suggested that as the craze grows it produces an offset at the surface of the specimen, and when the offset reaches a critical size, the specimen will then fracture in the craze.

The above theory can be made quantitatively by following eq. (13).

d p = (AD/J)(de/d)

2110 N. BROWN

He > Fast 2

(a 1 (b)

Fig. 7. Blunting of a surface flaw by a small craze.

where 6 is the total strain.

The craze fracture stress varies inversely with p , but the exact relations between craze fracture stress and size of craze has yet to be established. However, the above equation shows that the craze fracture stress decreases with total strain, increases with the strain rate, and increases as the diffusion coefficient decreases. Equation (23) says that polymers in which N2 diffuses slowly will tend to have greater craze fracture stresses for a given strain rate. It has been observed that the fracture stress of nylon in N2 at relatively low strain rates compared to other polymers is greater than in He. This observation is attributed t o the fact's that the diffusion coefficient of N2 is lower in nylon than in all the other polymers thus far tested.

The above ideas suggest that the low-temperature fracture stress of polymers in He or vacuum may be increased by a prior blunting of surface flaws. If the specimen is preconditioned by stressing in N2 or Ar at. a high strain rate for a short time so that the original flaws are blunted and the crazes do not grow very large, then the subsequent fracture stress in He or under vacuum could be increased beyond its original value.

In summary, small crazes increase the fracture stress by blunting the original surface flaws, but the craze fracture stress will decrease as the length of craze increases. High strain rates, low diffusion coefficients, and short testing times favor high craze fracture stresses.

CONCLUSION

A quantitative theory has been presented which explains the main features of environmental crazing at low temperatures. Many aspects of the theory are applicable to environmental crazing at high temperatures where environments other than N2 and Ar are effective. However, crazing at high temperatures will involve more bulk-shear yielding along with the craze yielding, so that the theory is not expected to be as successful at higher


temperatures. At very low temperatures, namely where the crazing agent freezes, then environmental crazing is not expected to take place. A material is not expected to be an effective crazing agent if it cannot wet and penetrate the polymer, as pointed out by Narisawa28 and the above theory.

My thanks to S. Fischer for the use of his unpublished research on nylon. The work was supported by the Advanced Research Projects Agency, the National Science Founda- tion, and the Army Research Office, Durham.

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7. A. N. Gent, J. Mat. Sci., 5,925 (1970). 8. S. Rabinowita and P. Beardmore, Critical Rws. Macromol. Sci., 1,1(1972). 9. E. Baer, paper presented a t Japan-U.S. Joint Polymer Seminar, Case-Western

165 (1970).

Reserve, October 1972; J. Macromol. Sci., in press. 10. D. Graham, J. Phys. C'hem., 66,1815 (1962). 11. D. Graham, J. Phys. Chem., 68,2188 (1964). 12. D. C. Braught, D. D. Bruning, and J. J. Schols, J. Colloid Interface Sci., 31, 263

13. S. S. Sternstein, paper presented a t Japan-U.S. Joint Polymer Seminar, Case-

14. R. P. Kambour and A. S. Holick, paper presented at American Chemical Society

15. P. Behan, M. Bevis, and D. Hull, Phil. Mag., 24,1267 (1971 ). 16. J. Murray and D. Hull, J. Polym. Sci. A-B,8,1521 (1970). 17. J. A. Sauer and C. C. Hsiao, Trans. ASME, 75,895 (1953). 18. S. B. Newman and I. Wolock, J . Res. Nat. Bur. Stand., 58,339, (1957). 19. 0. K. Spurr, Jr., and W. D. Niegisch, J. Appl. Polym. Sci., 6,585 (1962). 20. R. P. Kambour, Polymer, 5,143 (1964). 21. M. J. Doyle, A. Maranci, E. Orowan, and S. T. Stork, Proc. Roy. SOC. (London),

22. N. Brown, Mat. Sci. Ens., 8,69 (1971). 23. A. C. Knight, J. Polym. Sci. A, 3,1845 (1965). 24. V. Stannett, Difusion in Polymers, J. Crank and G. S. Park, Eds., Academic

25. B. Rosen, in Fundamental Phenomena in the Material Sciences, Vol. 4, L. J. Bonus,

26. W. G. Johnston and J. J. Gilman, J. Appl. Phys., 33,2716 (1962). 27. C. A. Kumins, in Difuswn in Polymers, J. Crank and G. S. Park, Eds., Academic

28. I. Narisawa, J. Polym. Sci. A-B,10,1789 (1972).

(1969).

Western Reserve, October 1972; J. Mucromol. Sci., in press.

Meeting, 1969; Polym. Preprints, 10,1182 (1969).

329A, 137 (1972).

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Press, New York, p. 113.

Received March 21, 1973 Revised May 21, 1973

a theory for environmental craze yielding of polymers at low temperatures

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