nucleation and growth of crazes in amorphous polychlorotrifluoroethylene in liquid nitrogen

JOURNAL OF POLYMER SCIENCE Polymer Physics Edition VOL. 13,1315-1331 (1975)

Nucleation and Growth of Crazes in Amorphous Polychlorotrifluoroethylene in Liquid Nitrogen

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

University of Pennsylvania, Philadelphia, Pennsylvania 191 74

Synopsis The density of mature crazes initially increases linearly with stress and then more

rapidly at higher stresses. Once the crazes become observable then density was independent of time. The lowest stress at which an appreciable density of crazes was produced corresponds to the proportional limit. The average velocity of mature crazes was constant for a given stress and varied exponentially with the stress. The velocity depended on stress in the same way that the post-yield point stress depended on strain rate, whereas the yield point varied differently being a nonlinear function of the logarithm of the strain rate.

The density of crazes was quantitatively related to the concentration of surface defects a t which the crazes nucleate. The craze velocity was directly related to the diffusion coefficient of N, into the polymer. The analysis indicates that bulk diffusion of the Ni governs the craze velocity and that plasticization of the tip of the craze is most important for the nucleation and growth of a craze in PCTFE.

INTRODUCTION

Up to the present time the phenomenon of crazing at low temperatures in gaseous environments has been studied by observing the effects on the macroscopic stress-strain curve as shown in Figure 1. The typical effects from a gas such as Nz or Ar is to produce an upper yield point and to cause the strain rate to profoundly change the shape of the curve and the magnitude of the yield point. Hoare and Hull' and Brown2 have pointed out that the shape of the strcss-strain curve depends on the kinetics of nucleation and growth of the crazes. The following equation by Brown2 was shown to be fundamcntal in determining the stress-strain curve for craze yielding :

de,/dt = pbf l dl/dt (1)

where de,/dt is the strain rate produced by crazing; p is the density of crazes; b is the displacement produced by a craze; f is a shape factor; 1 is the length of the craze, and dl/dt is the velocity of the craze. It is the pur- pose of this papcr to determine the physical basis of the craze density and velocity and to correlate these with the effect of the strain rate on the stress-strain curve.

The previous studies of the nucleation and growth of crazes have been conducted a t room temperature and have been reviewed by K a m b ~ u r . ~ These studies indicated that the growth could vary linearly or logarith-

1315

@ 1975 by John Wiley & Sons, Inc.

1316 BROWN AND FISCHER

32r 28 t / OD2 24 -

QUENCHED

0 4 8 12 16 20 24 28 STRAIN (%)

Fig. 1. Strain-stress curves of quench PCTFE at 78°K in Na and He environments at various strain rates (min-1).

mically with time. The nucleation of crazes was found to be both stress- and time-dependent. The nucleation of crazes in the vicinity of room temperature seems to agree with the concepts of thermal activation in that nucleation takes a longer time the lower the stress and the crazing stress decreases with increasing temperature. These observations may not be applicable to gaseous crazing at low temperatures where it has been ob- served2v4* that above a critical temperature crazing is no longer observed. Also studies of nucleation and growth at the higher temperature range is more complex because concomitant shear flow may be an important factor, as pointed out by Kambour and Robert~on.~

In this investigation the density and size of crazes were measured as a function of time under a constant stress as well as a function of the stress. The effect of strain rate on both the yield point and the stress beyond the yield point was measured and compared with the effect of the stress on the velocity of the individual crazes. The observed craze density was analyzed in terms of the density of defects at which the crazes were nucleated. The velocity was also quantitatively related to the diffusion of the gas into the craze.

EXPERIMENTAL

Material

Polychlorotrifluorethylene was chosen as the material because this investigation was part of a larger study on the effect of crystallinity on

CRAZES IN PCTFE 1317

crazing. The PCTFE has the commercial designation Kel-I? 81 Grade 3 by the 3M Company. It has a molecular weight of approximately 1600 (CF&FCl) units. It is considered to be the purest form of PCTFE that is manufactured by 3M. It was received from the Fluorocarbon Co&- pany as 0.035-in. compression-molded sheet and is called LOX-Grade which means it was uniformly quenched in water from about 300°C. The as-received material was made into tensile specimens with a uniform gage length of .5 in. and a width of 0.200 in.

Test Procedures

The tensile specimens were deformed in liquid nitrogen with an Instron machine. The stresses as reported are within f0.2 ksi. The strains were obtained from the crosshead motion of the machine and their absolute values are uncertain by about flOyo in the plastic region and about f50Y0 in the low strain elastic region because the effective gage length is a function of the strain.

While the specimen was immersed in liquid nitrogen it was loaded in the Instron machine to a given load within about 30 sec and the load was held constant to within =t=O.5% for a certain time. The specimen was removed from the tensile machine in an alignment jig and then the crazes were observed under a microscope and photographed at room temperature. The same specimen was retested in liquid nitrogen at the same load for another period of time. The procedure was repeated until the specimen fractured. In some experiments the craze density was determined after the specimen was brought to the vicinity of the yield point under a constant strain rate and then unloaded.

Photographs were taken a t various magnifications in order to take into account the effect of magnification on the measurements. The crazcs were observed under transmitted and reflected light. Usually transmitted light gave the best contrast. The lengths of individual crazes were measured from the photographs and only the crazes which did not reach the edge of the specimen were measured. The uncertainty in length of the craze at the magnification of 50X was about fO.01 mm, being somewhat sensitive to the contrast to the photograph. The measured density of crazes was averaged over two fields of view which were chosen randomly from within the gage length of the specimen. The field of view was about 0.01 cm2 de- pending on the density. The density could vary by as much as a factor of two for different fields of view. The counts by different people varied only about f 10%. The greatest uncertainties in counting occurred at the be- ginning when the crazes were just wide enough to see and later when the density was high and the crazes were long so that overlapping occurred. The measured density increased somewhat with magnification, but the factor was less than the density fluctuations that were observed in different areas.

The change in stress with strain rate was measured beyond the yield point by suddenly changing the strain rate from a low to high value and back to the low value again. Thus, the effect of the change in the stress with strain could be minimized in order to obtain the strain rate sensitivity (b In k//da),.


EXPERIMENT& RESULTS

Stress-Strain Curves

The stress-strain curves at various strain rates a t 78°K in liquid nitrogen and helium environments are shown in Figure 1. In helium, crazing was not observed and the effect of strain rate is very small. In liquid nitrogen crazing was first observed in the vicinity of the proportional limit. Gen- erally, the density of crazes increases with the strain rate as in the case of other polymers.' A major question to be answered is whether the increase in density was caused by the increase in strain rate per se or by the higher stress, since the yield point increases with strain rate.

Density of Crazes

The density of crazes is defined as the number of crazes per unit area. The crazes were nucleated on the surface, often at visible defects, such as a notch, scratch, void, or embedded particle of dirt. The density of crazes was usually higher at the machined edges of the specimen with the highest density being at the points on the edge near the shoulder of the specimen where the stress concentration was greatest. There seems to be little doubt that the probability of observing a craze at a particular point on the specimen is directly related to the magnitude of the local stress.

In the constant-stress experiments a certain time elapsed before crazes were visible. In order for a craze to be seen for a given set of illumination conditions and at a certain magnification, it had to achieve a certain critical thickness. Once the craze reached the critical thickness, it grew with what appeared to be a constant thickness and with a tip of constant shape; such a craze is called a mature one. Figure 2 shows the growth of crazes. An important observation was that all the crazes in a particular region reached their critical thickness at about the same time, and when they began to grow as visible crazes, the density remained constant. When the crazes became very long and especially a t high craze densities, they would ac- cidentally join along their lengths and cause small apparent decrease in density. Figure 3 shows the density of crazes measured at 50 magnification under reflected light as a function of stress for times beyond that required to incubate mature crazes. The density was measured in areas which do not include the edge of the specimen. The relationship between stress, u, and density, p, is given by

p = C ( u - uo) crazes/cm2 (2) where C = 1200 crazes/cm2-ksi and uo = 13.0 psi. uo is based on a linear extrapolation of the data to zero density. However, the lowest stress a t which crazes were actually observed was limited by the length of the incubation time which, for a stress of 13.5 ksi, was about 2 hr a t a magnification of 50 X.

Additional density measurements were madc on specimens which were loaded just beyond the yield point a t a constant strain rate. The strain in the specimen was controlled so that the time under the maximum stress was greater than the incubation time but was not too long in order to keep the craze length short enough for the density to be conveniently measured.


Fig. 2; Crazes growing under a constant stress of 14 h i at 20,50, and 80 minutes, 50 X magnification.

STRESS ( K S I )

Fig. 3. Density of crazes vs. stress measured at 50 X magnification. Stress wat~ stant during growth.

con-


1 I I

STRESS (KSI) Fig. 4. Density of crazes vs. stress.

These measurements wera donc at higher stresses than thc previous ones so that i t was necessary to photograph the crazes at higher magnifications in order to resolve them. The magnifications ranged from 150X to 250X and as mentioned in the Experimental section, the apparent variation of density with magnification was less than the fluctuations obserCed in different areas. All the density data including that from Figure 3 are plotted in Figure 4, which shows two major points: (1) the rate of change of density with stress increases with the stress, and (2) the density becomes very small below the stress of 13 hi. The linear relationship given by eq. (2) is only applicable over a narrow range of stress in the low-stress region. Figure 4 represents the density for times beyond the incubation period where the density depends on the maximum stress to which the specimen had been exposed.

Growth Rate of Crazes

During thc incubation period a craze reached an equilibrium thickness and then grew at a constant thickness. The curves of length versus time are shown in Figures 5a and 5b for stresses of 13.5 and 15 ksi. The incubation time was about 2 hr for 13.5 ksi and 1 min for 15.0 ksi. Although the initial lengths of the crazes at a given stress might vary by a factor of two, the subsequent growth rate was essentially linear and about the same for each craze. The crazes whose growth was measured were isolated crazes. Thc growth of crazes which grew too closely together were not measured because they would interact. When the tips of two growing crazes were

CRAZES IN PCTFE

1.4

1.2

- $ 1.0 - I I- 0.0 W z W -I

0.6

0.4

0.2

1321

-

-

-

-

-

-

-

PCTFE QUENCHED

0 :

0

0

I I I

I I I I I

~ = 1 3 . 5 K S l

01 I 1 I I I I 0 2 4 6 8 10 12

TIME (HRS)

(4

PCTFE QUENCHED u - 15.0 K S I

0.0

E - 0.6

close together, the crazes would join by pointing toward each other away from the normal growth direction which is perpendicular to the applied tensile stress. At fracture, crazes often extended across the entire width of the specimen. The growth of isolated crazes could be observed up to a length of 1.0 mm at 13.5 ksi where the density was low and up to lengths of only 0.7 mm at 15.0 ksi where the density was high.

1322

1.4-

1.2-

1.0

A

5 I 0.8 - I I-

W J

: 0.6

0.4

0.2

0

BROWN AND FISCHER

0 = 13.5 KSI 0

A = 14.0 K S I

0 = 15.0 K S I 0 a 14.5 K S I

- d

-

-

-

-

I I I 1 37 38 39 40

o.2- 0 0 5 10 15 20 25 30

(MIN.) , ( K S I ) + ,2.5Cx 10-16

Fig. 7. Curve in Fig. 6 replotted BS length vs. te2*&.

Growth was observed under a series of stresses from 13.5 to 15.0 ksi. The natural 1ogarit.hm of the time to obtain a given length of craze was plotted against the stress (Fig. 8). A set of parallel lines was obtained whose slope was 2.5 ksi-I. The average growth from data as in Figure 5 was plotted against a stress-compensated logarithm of the time as shown in Figure 6. Then the curve in Figure 6 was replotted in Yigure 7 as a function of te2,& with the result that an average of all the growth dat.a could be represented by the following equation:

( I - l o ) = KPSU (3)


where K , the rate constant, is 2.64 X 10-l8 mm/min and l o = 0.24 mm. The foregoing value of l o is an average value of the craze length after the incubation period. A more detailed analysis of the data indicated that lo increases with the stress in agreement with the observations by Sauer and Hsiao.8 The rate of lengthening during the incubation period is greater than that indicated by eq. (3), as shown by the very existence of the term lo and by the extrapolation of a growth curve to zero time (Fig. 5b).

At the present time there are no data as to how much the growth rate was affected by the intermittent loading that was required to bring the specimen to room temperature in order that it be photographed. There were no in- dications that these interruptions caused any discontinuities in the craze velocity. Equation (3) is viewed as being an average representation of the growth for the approximately 25 individual crazes that were observed in the stress range of 13.5-15.0 ksi.

Strain-Rate Effects

As shown in Figure 1, the strain rate has a profound effect on the shape of the stress-strain in a nitrogen environment where crazing is the primary mode of plastic deformation. The effect of strain rate on the yield point is shown in Figure 8. Contrary to prior observations with other polymer~,~.gln i versus uy, the yield point is not linear. The effect of strain rate

CI - 1.0 z f

t I-

- B

2 a i? I

a (3

0.1

V

W

I- J

0 5 0.01

a P-

a 0 W

0.001 12 14 16 18 20 22 24

STRESS ( K S I )

Fig. 8. Various effects of stress o I rate of crazing. Time for a craze to grow to a Change in postyield point stress with strain rate. Change in certain length vs. stress.

yield point with strain rate.


0.04 m i d I 0.004 b.4 0.004 bO4IO.004 b.4

EXTENSION

Fig. 9. Strain rate change experiments to determine effect of strain rate on the postyield point stress. The range for a given strain rate is indicated above the curve.

on the stress beyond the yield point was obtained from strain rate change experiments shown in Figure 9. In addition the change in thc logarithm of the growth rate of crazes versus stress is also plotted in Figure 8 where the average slope is 2.5 ksi-l as indicated in the previous section. Two features of Figure 8 are important: (1) The slope far the postyield point stress is the same as that for craze growth, and (2) the slope for the yield paint is nonlinear but becomes parallel to the other curves at low strain rates. If the slope for the yield point is not the same as that for the craze velocity, then it may be concluded that another dynamic process in addition to the craze velocity contributes to the yield point. Most likely, the increase in craze density with stress represents the additional dynamic factor.

ANALYSIS OF RESULTS

Nucleation of Crazes

The stress exerts the primary role in determining whether a craze forms. Time plays a part in determining when the craze can first be observed. The magnitude of the applied stress to produce a craze depends on intrinsic properties of the polymer, the defects at the surface of the specimen, and the environment.

The Incubation Period. A craze is nucleated a t a point in the specimen by a certain value of the stress. During the incubation period the craze thickens and lengthens until it is thick enough to be visible. For a given stress, all the crazes reach maturity a t about the same time as evidenced by the fact that the density of crazes becomes constant during their subsequent growth. Thus, there is no indication that there is a continual nucleation wherein the nucleation process is thcrmally activated in concert with the stress. The role of time is simply part of the growth processes and not part of the conditions for nucleation. It takes longer to see a craze at a low stress simply because the growth rate is a function of stress. Thermal activation may be significant for crazing at higher temperatures.

Connection Between Surface Defects and Craze Density. The direct observations show that crazes nucleate at points of stress concentration

First the role of time will be discussed.


which occur at the surface of the specimen. This fact was also emphasized by Sauer and Hsiaos and has become an integral part of most theories on craze nucleation. Experiments which we have conducted show that abrad- ing the specimen decreases both the proportional limit and the yield point. We shall show how the density of defects is related to the craze density. The,applied stress which nucleates a craze is given by

uc = .co /q (4)

where uc is the stress to nucleate a craze in the absence of a surface defect and q is the stress concentration factor of the defect. If eq. (4) is combined with the experimental eq. (2), then

p = cuco[; - 3 Qmax

(5 )

where Cis a constant and equal to 1200 crazes per ksi-cm2 for this particular polymer and qmax is the stress concentration factor for the most severe defect in the specimen. The physical significance of C can now be quantitatively interpreted. If it is assumed that one craze nucleates a t a defect

The factor dpD/d(l/q) is the change in defect concentration per unit change in l/q; this factor depends on the surface condition of the specimen. It is also shown that the craze density varies inversely with uco and it increases as qmaX increases.

dPD CUc0 dq qZ

The result says that if C is a constant the change in density of defects per unit change in q varies as - l/q2. Since q generally decreases as the size of the defects decreases, eq. (7) also says that the concentration of defects varies inversely as the square of their size. The experimental data in Figure 4 show that at high stresses, C increases with stress so that the concentration of defects increases as q decreases a t a faster rate than indicated by eq. (7). These results are consistent with the general observations of defects in solids, namely, the concentration of small defects is greater than that of larger ones.

Physical Basis of uco. Crazes nucleate a t the stress concentration produced by surface defects and are perpetuated by the stress concentration produced by the craze itself. Let us now consider the other physical parameters that determine whethcr craze yielding or shear yielding occurs. The yield point for shear yielding is not sensitive t p defects which cause stress concentrations. Parrish and Brown'O showed that notches made with a razor blade had practically no effect on the magnitude of the yield point in poly(ethy1ene terephthalate). The bulk polymer undergoes a rather homogeneous deformation prior to the shear point, whereas the nucleation and growth of a craze involves a highly localized form of deformation at the point of a highly localized stress concentration. We shall

It is also noted that

(7) - = - -


describe the factors, beside the stress concentration itself, which determine the stress, ueo, which causes crazing.

The initiation of the craze involves the enlargement of submicroscopic voids ia the polymer. Thus, there are two factors which determine uCo:

(1) the stress to produce the additional surface area as the void enlarges, and (2) the stress to produce the localized plastic deformation that is also required to enlarge the void and that leads to the fibrillation that is char- acteristic of a craze. These two factors, the surface effect and the plastic effect, are described by the following equation which was proposed by Andrews and Bevan" and modified by Brown2

Go = 3 [(2YS*/1') + (4.3%43,) 1 (8)

where y is surface energy, T is the radius of the submicroscopic voids; uY is the yield point for shear deformation; & and p,, are factors by which the N2 environment reduces the surface energy and reduces a, by plasticization, respectively. The factor 4.3 occurs because the void enlargement process involves the localized nonhomogeneous plastic deformation which is comparable to the enlargement of a void in a solid under a hydrostatic tension. The best known values of the above parameters are y = 50 erg/cm2, uY = 29 ksi from Figure 1, and PS = 1/2 from data on the adsorption iso- therm~. '~-" Nothing is known about except that i t is related to pB in that both involve the interaction between N2 and the polymer. Since pp varies from zero to one it will be assumed to equal PB.

The value of T is a matter of speculation based on what is known about the morphology of polymers in the amorphous regions where the largest value of T is expected to occur. The microscopic observations of YehI6 indicate that T is not appreciably greater than 20 A, which is roughly the limit of resolution. A lower limit for T can be based on the observations of the voids that are observed where hard models of polymer molecules are randomly packed. If the diameter of the molecular chain is approximately 6 A then the volume of randomly packed chains would frequently have voids that are a t least 2 A in size. The crazing would start at the largest microvoids in the system. Thus, if r is between 2 and 20 d, uco is between 300-200 ksi, respectively. Also the major part of the stress is associated with the plastic deformation and not the formation of new surface areas, in agreement with the calculation by Kambour.

The above calculation also indicates there must be defects which will produce stress concentrations as large as 14-23 in order that crazes be nucleated at the observed stress of 13 ksi. Equation (8) is a rather simpli- fied model for trying to calculate the absolute value of the stress for crazing, but i t does agree qualitatively with the experimental observations that the stress for crazing increases with yield pointe and the fact that a reduction in the concentration of N2 also increases the stress for crazing.

Craze Velocity

Since the N, causes crazing i t is assumed that the craze velocity is controlled by the flux of Nz into the tip of the craze. According to eq. (3), the craze velocity is constant for a given applied stress. Thus, the craze grows with its tip containing a certain concentration of NZ which plasticizes


the polymer in order that the craze can grow under the applied stress. Thus, under conditions of constant velocity the craze velocity may be described as follows:

dl /d t = A X drift velocity (9) where A is related to the surface coverage of the N2 on the inner surface of the craze and the drift velocity of the NT, is given by

drift velocity = Jve-(Q-u"/kT) (10) J is the jump distance for the N2 molecule; Y is the vibrational frequency, Q is the activation energy for diffusion, u is the stress, and v is the activation volume.

(11)

dl /d t = A D / J (12)

In terms of the diffusion coefficient D = J 2 y e - ( Q - f f V / k T ) = D ( . = o),,uv/kT

Combining eqs. (9), (lo), and (11)

This equation had been presented previously* without being derived. A , the surface coverage, varies from zero to one and is related to the adsorption coefficient.

During the thermally activated process, it is visualized that a certain length of the polymer molecule undergoes an undulation which permits the N2 molecule to move a distance, J . The undulation would involve a molecule length, J , and displacement equal to the diameter of the N2 molecule, d,. If the width of the polymer molecule is 4, then

Now the activation volume, v , may be simply interpreted.

v = J&d, (13) Another way of interpreting v is to say that during a thermally activated diffusion process, the stress acts on a surface Jdp while it undergoes a displacement d,. From eq. (3) v/7Sk = 2.5 ksi-' and v = 390 A3. From eq. (13) and using the following values of the van der Waals' diameters: d, = 3.7 A and dp = 6 A, J = 18 A. This value of J is consistent with theoriesl6 of diffusion of small gas molecules in polymers.

Now it is possible to give a qualitative interpretation of the physical parameters which make up the experimental rate constant K in eq. (3). Combining eqs. (3), ( l l ) , and (12),

It is now possible to calculate J .

AD(u = 0) J

K =

D(u = 0) , the diffusion coefficient under zero stress, can now be calculated from eq. (14). At 78°K A = 1, and using the above value of J and the experimental value of K , the diffusion coefficient of N2 in quenched Kel-F a t 7S"K, D ( u = 0) = S X 10-28 cm2/sec.

It is now of great interest to determine how this last result compares with t i e actual bulk diffusion coefficient at 7S"K. There are no direct measure- m:nts of diffusion at 7S"Ii, and the best that can be done is to extrapolate room-temperature measurements to 7S"K. Unfortunately, there are only


permeability measurements of Nt in PCTFE in the neighborhood of room temperature” which gives an activation energy for the permeability of 12 kcal/mole. It is estimated from experiments on another polymer’8 that the enthalpy for sorption is between -3 and -4 kcal/mole. Thus, it is estimated that the diffusion coefficient of N 2 in PCTFE is given by b = O.le-Q’kT where Q = 8-9 kcal/mole and at 78”I<, D(u = 0) = 10-23 to

This result is comparable to the extrapolated value of the diffusion coefficient for polycarbonate based on actual diffusion data.’8 Thus, the value obtained from eq. (14), D(u = 0) = cm2/sec, is not unreasonable.

Based on the fact that the diffusion .coefficient as calculated from eq. (14) is nearly the same as that estimated for the bulk diffusion coefficient, we conclude that craze growth is controlled by the diffusion of Nz into the bulk polymer. If bulk diffusion governs the craze velocity then the N 2 enters the bulk in order that the polymer can be plasticized to the extent necessary for the continuation of craze growth under the applied stress. It is unfortunate that there are no direct data on the effect of stress on the diffusion coefficient in order that the theory can be further checked by other types of experimental observations. In a prior paper,2 D ( u = 0) was calculated from a more complex theory using strain rate data instead of craze velocity data. The connection between strain rate effects on the yield point and craze velocity data is not simple, and thus it is not surprising that the previously estimated value of D ( u = 0) was different than the present results. Also the previous value of D(u = 0) led to the conclusion that the N 2 acted mostly a t the surface of the craze rather than within the bulk polymer and that the primary action of the N 2 was to reduce the surface energy rather than to plasticize the tip of the craze. The present results indicate that plasticization is more important. However the critical depth of plasticization, that is the size of the plasticized tip, may only be about 18 A, which is the value of J . In order to answer this last point more definitely it is necessary to know what concentration of N2 is required for growth under a given applied stress.

Marshall, Culver, and Williams19 studied the craze velocity in liquid environments. In their model the craze velocity was governed by the flux of liquid as determined by the pressure gradient between the tip of the craze and the surface of the specimen (end flow). Our model assum= that the flux of N2 from the surface occurs by gaseous diffusion and therefore the flux of N2 from the surface to the tip of the craze is extremely rapid compared to diffusion into the polymer itself. Thus, our model leads to a constant velocity and the Marshall, Culver, and Williams model leads to t-”’ dependence for the velocity. Marshall, Culver, and Williams also considered a model governed by diffusion of the liquid along the edge of the craze front from the side (side flow). The side flow process gives a constant craze velocity after the initial period of growth.

The general size of the interconnected voids in a fully developed craze according to the observations of Kambour and Holick” and Behan, Bevis, and Hull2‘ is about 100 A. Since the size of the NZ molecule is 3.7 A, the transit time from the surface of the specimen to the tip of the craze should be very short. A model of the N f distribution at the tip of the craze is

cm2/sec.


Fig. 10. A model of the N, distribution at the tip of a growing craze. The arrows indicate the local stress. The NI molecules are the open circkes.

shown in Figure 10 under the conditions where bulk diffusion controls the craze velocity.

Strain Rate EfFed

The strain rate sensitivity, (a In i / ? l ~ ) ~ , is a basic parameter for inter- However, the interpretation is preting the mechanism of deformation.

usually based on the assumption that

(15) 6 = Cenv/kT

where C is a constant, independent of u, and thus, if v is a constant, (a In ;/?la), would be constant. However, in the case of the yield stress, C is not a constant because according to eq. (1) and the results of the present investigation

i, = pbjlKe2*5by (16)

where p and I, the density and length of craze, respectively, are functions of uy) the yield stress. Thcrefore, it is not expected that generally cry versus d would be linear in agreement with Figure 8. The fact that uy is often observed to be a linear function of a, may be attributed to the limited range of i that is usually observed. It is also noted in Figure 8 that the strain rate sensitivity of the postyield point stress is associated with the same activation volume as the craze velocity. This observation is well ex- plained by the fact that p and 1 are constant during the postyield point dynamic strain rate change experiments. The fact that the activation volume from yield point measurements approaches that for the craze velocity a t low strain rates is attributed to the observation that the changes in p and I with stress are much smaller a t low strain rates. Figure 1 shows that the difference between the yield point and the proportional limit, the stress where crazes are nucleated, decreases with strain rate. Therefore the variation in p and 1 with stress approaches zero as the applied strain rate approaches zero.

No quantitative observations were made on the effect of stress on b, the craze displacement parallel to the stress. It is likely that b would also be a function of the stress, in which case the activation volume from yield point measurements would tend to depart further from the value obtained from the craze velocity. In order to gain the most fundamental under-


standing of craze yielding, it is suggested that measurements be made of the velocity of individual crazes under a constant stress.

CONCLUSIONS

The following conclusions are applicable to low temperature environ- mental crazing:

1. The density of crazes is a function of the applied stress and the density of surface defects which produce a definite stress concentration.

2. The craze velocity of mature crazes is, on the average, constant for a fixed stress and varies exponentially with stress.

3. The velocity of craze lengthening is more rapid during the incubation period than thereafter.

4. The craze velocity is governed by the surface concentration of ad- sorbed gas and by the diffusion of the gas into the polymer.

6. The gas lowers the stress to produce a craze by plasticizing the tip of the craze and by reducing the surface energy with the plasticizing effect being more important. 6. 7.

The effect of In i on the craze yield point is not linear. The activation volume from measurements of craze velocity versus

stress is the same as that for In i versus the postyield point stress but not equal to that for In i versus the yield point.

The research was uupported by a grant from the U.S. Army Research Office-Durham. The National Science Foundation Grant DMR72-03025 whicR supports the Materials Research laboratories provided a stipend for S.F. and the use of the Central Laboratory Facilities.

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Received January 8, 1975 Revised February 24, 1975

nucleation and growth of crazes in amorphous polychlorotrifluoroethylene in liquid nitrogen

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