an equation for creep in terms of craze parameters

An Equation for Creep in Terms of Craze Parameters

NORMAN BROWN, BRUCE D. METZGER, and YASUFUMI IMAI,* Department of Metallurgy and Materials Science and the Laboratory for

Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104

Synopsis

The creep curves of poly(chlorotrifluoroethy1ene) (PCTFE) and poly(methy1 methacrylate) (PMMA) were measured at temperatures near 78OK in nitrogen environments so that practically all the creep strain was produced by crazing. Data were obtained over a range of temperatures, stresses, and partial pressures of the nitrogen. It was found that all creep curves had the form, c1I3 = and s is a constant which depends on the conditions of the test. The microparameters such as number, dimensions, and velocity of the crazes were measured in each specimen. A theoretical creep equation was derived in terms of these microparameters. There was good quantitative agreement between theory and the experiments.

+ st , where €0 is the initial strain

INTRODUCTION

We have not been able to find any experimental investigations of creep where it was demonstrated that crazing was the only form of deformation. There are many investigations of creep in polymers where shear flow occurs. Verheul- pen-Heymans and Bauwens1I2 studied crazing and its relationship to creep in polycarbonate at and above room temperature, but under these conditions shear flow is also a very prominent mode of deformation. Brown3 presented a theory where it was predicted that the square root of the creep strain should vary linearly with time for pure crazing. The major goal of this investigation is to determine the creep equation that governs craze deformation and to correlate the equation with the microparameters of the individual crazes.

Shear flow was completely eliminated by working with the polymers poly- (chlorotrifluoroethylene) (PCTFE) and poly(methy1 methacrylate) (PMMA), at low temperatures in the neighborhood of '78'K. Crazing was produced by nitrogen environments as described in previous investigation^.^-^ Many creep curves were produced under various combination of stress, temperature, and gas pressure. It was found that all the creep curves could be described by the equation $I3 = + st, where &,I3 and s are constants which depend on the microparameters of crazing. The microparameters that determine the creep equation are craze density, defined as the number of crazes per unit area of the specimen, dimensions of the craze, and the velocity. These parameters were measured microscopically.

The basic equation which relates the microproperties of crazes to the macroscopic deformation is analogous to the well-known Orowan equation which relates the properties of dislocations to the macroscopic strain in crystalline solids. In the Orowan equation, the strain t is given by

c = p b A

* Present address: Nagasaki University, Nagasaki, Japan.

Journal of Polymer Science: Polymer Physics Edition, Vol. 16,1085-1100 (1978) 0 1978 John Wiley & Sons, Inc. 0098-1273/78/0016-1085$01.00

1086 BROWN, METZGER, AND IMAI

where p is the density of dislocations sources per unit volume, b is the Burgers vector, and A is the average area swept out by a dislocation. A notable use of this equation has been made by Johnston and Gilman7 who first calculated stress-strain curves. The Orowan equation is used extensively to correlate macroscopic deformation and the microstructure of dislocations. Hoare and Hull8 originally proposed an analogous equation for crazes. Curran et al.9 used a similar equation to equate the macroscopic behavior of materials to the nucleation and growth of shear bands that form under unstable adiabatic conditions.

The Hoare-Hull8 equation with modifications by Brownlo is given by

where t is the craze strain p is the number of crazes per unit area of the specimen surface, y is the ratio of surface of the specimen to its volume, b is the craze opening displacement, and A is the average area of a craze. The density of crazes is given in terms of area rather than volume as in the Orowan equation because crazes in homopolymers generally nucleate a t the surface of the specimen in contrast to dislocations which generally nucleate throughout the volume. An- other important difference between eq. (1) and the Orowan equation is that b, the Burgers vector or the displacement vector, is a constant since it is a vector in the crystal lattice, whereas b, the craze opening displacement, is not the same for all crazes and may vary with time. Thus, b in eq. (1) is an average value which is averaged over the area of the craze.

THEORY

The Creep Equation

Equation (l), which is the basis of our theory, refers to a static situation and is fundamentally geometric. The craze is taken to be a plate-like structure, as illustrated in Figure l(a), whose area is determined by two dimensions: C, par- allel to the surface of the specimen, and p , perpendicular to the surface. Since the crazes are approximately semielliptical in shape, the area of a craze is given by

A = a p C

p is now the average dimension of the craze in the direction perpendicular to the axis L. a, the shape factor, is approximately equal to 1 since a equals 1 if the craze were exactly semielliptical. Since it is observed that the area A is u-sually perpendicular to the tensile strain and since b is also perpendicular to the area of the craze, the strain is simply proportional to bA. Thus, eq. (1) becomes

t = p y b a p C (3)

In order to obtain the creep equation from eq. (3) it is necessary to make as- sumptions concerning the time dependence of the four microparameters p, b, p , and C .

CREEP EQUATION 1087

ORIGINAL EXTENDED RETRACTED POLYMER CRAZE CRAZE

(b)

Fig. 1. Models of a craze showing the microparameters. u is the direction of tensile stress.

In the theory p is assumed to be a constant. Brown and Fischerl' showed that, for PCTFE crazes in liquid nitrogen, the density of crazes reached a constant value very shortly after the load was applied. The density of crazes depends on the stress. The theory of craze nucleation by Argon et a1.12 predicts a time- dependent nucleation which involves a rapid approach to saturation after the stress is applied. The experimental observations of Argon et al.12 on polystyrene (PS) at room temperature showed a transient in craze nucleation, but the transient time is generally small compared to the period of craze growth. We, therefore, neglect any time dependence of p in this theory.

Our model for craze growth assumes that the craze maintains its shape so that there is always a constant proportionality between b, p , and 4. Thus let

where kl and k2 are constants. In a previous theory3J0J1 it was assumed that b was constant. However, experimental observations13J4 show that b increases as the area of the craze increases. The present assumption is made more plau- sible when it is realized that in the limit of a weak craze (i.e., a crack), b varies linearly with l. If b varies along the length of the craze, then the average value of b would be used in eq. (3).

The next assumption concerns the growth velocity. Experimental observa- tions11J2J5-17 of craze growth usually indicate that the crazes grow with a con-


stant velocity. It has been showtl by Sauer and Hsiao15 and Argon et a1.12 for PS at room temperature, and by Brown and Fischerl' for PCTFE at liquid-nitrogen temperature that

1 = ( lo + u t ) (5) where 10 is an initial length of the craze that forms during or shortly after loading and u is the craze velocity. Substituting eqs. (4) and (5) into eq. (3) gives

€ = p y a klkZ(l0 + ut)3 (6) Equation (6) is the predicted creep equation. Since every quantity in eq. ( 6 ) is measureable, it is possible to make a complete comparison with experiment. Equation (6) predicts

(7 ) €1/3 = 113 + st EO

where to, the strain at t = 0, is given by

E6I3 = ( p y a klkz)1/3~0

s = ( p y a klk2)1/3u

(8)

and s is given by

(9)

Relation of Craze Opening Displacement to Thickness

The theory involves the craze opening displacement b, but the experimental observation involves the thickness of the craze after it retracts. The question now arises as to the relationship between b and the observed thickness.

Let a0 be the primordial or original thickness of polymer from which the craze originated and a the thickness in the retracted state. The thickness of the extended craze is a0 + b [Fig. l(b)]. The measureable ratio of the macroscopic strain in the retracted to that in the extended state q , is given by

(10) q = (a - ao)/b

p = b/(ao + b )

Defining p as the fractional porosity in the extended state

(11)

Eliminating a0 from the above equations yields

(12)

Microscopic observations1s20 indicate that p = 0.5 for many crazes. For this porosity, b = a for q = 0 (complete retraction) and b = 0.5a for q = 1 (no retraction). The experimental results show q = 0.1 so that b = 0 . 9 ~ if p = 0.5.

U b =

q + (1 - pup

EXPERIMENTAL METHODS

The PCTFE was the same material used in a previous investigation on the nucleation and growth of c raze~.~J l The resin was Kel-F-81 made by the 3M Company. The specimens were made from 0.9 mm sheets which were com- pression molded and water quenched from 300°C by the Fluorocarbon Company

CREEP EQUATION 1089

and called LOX-Grade. The material was very reproducible in its mechanical behavior. The PMMA was commercial Plexiglass-G from Rohm and Haas in the form of 1 mm thick sheets. The creep specimens had a dumbbell shape with a 12.7 mm uniform gate length and 5 mm width.

The creep machine was a lever type with a 5:l load ratio. The elongation was measured with a Pickering LVDT transducer whose sensitivity was about 4 X

The specimen was enclosed in a copper environmental chamber which was cooled by liquid nitrogen. The temperature of the specimen was controlled within f 0.2"C by a heating wire which surrounded it. The partial pressure of the Nz was precisely maintained by mixing helium and Nz whose flow rates were determined by manometer-controlled flow meters. The variation in partial pressure was about f 1% during an experiment. The helium functions as an inert gas.

The density and length of the crazes were measured from optical micrographs (Fig. 2). The dimension of the craze perpendicular to the surface was obtained by metallographically polishing a longitudinal section of the specimen and ob- serving the crazes under the optical microscope (Fig. 3). The thickness of the crazes was measured by two methods: from a replica in the transmission electron microscope (TEM) and by the scanning electron microscope (SEM) after the specimen was coated with gold. Both methods gave the same results; the replica method [Fig 4(a)] gave greater resolution but it was slower than the SEM method [Fig. 4(b)].

The retraction experiments consisted in making two scratches on the specimen about 1 in. apart, and measuring the distance between the scratches with a traveling microscope on the virgin specimen and after the specimen was removed from the creep machine. The sensitivity of the traveling microscope was in. The total error in measuring the strain of the retracted state was f 0.01% which was very small compared to the creep strains of a few percent. Therefore, q was determined with a high accuracy.

mm. The transducer signal was recorded on a Varian x - t recorder.

EXPERIMENTAL RESULTS

Shape of the Creep Curves

The PCTFE and PMMA were tested in pure He, where no crazing could be observed. The creep rates were practically zero. The deformation was essen- tially elastic under the range of stress and temperature used in this investigation. These results are consistent with previous stress-strain experiments in He.4,5 Therefore, it is concluded that bulk deformation did not contribute significantly to the creep curves.

Typical creep curves for PCTFE in Nz environments are shown in Figure 5 for ranges of stress, temperature, and partial pressures of Nz. The short-time (Fig. 5) and the longer-time creep tests, up to 800 min duration, all have the same general shape in that the creep curve always accelerates in accord with our theory. The creep curves for PMMA also exhibited a continuous acceleration but their total strain to fracture was much less than for PCTFE.

The creep curves were replotted as €1'3 vs. t as shown in Figures 6 and 7. These curves are linear in accord with the theory.


(b)

Fig. 2. Micrographs for measuring p and t. (a) P C T F E 0.5 atm, 78'K, 130 MPa. (b) PMMA 1.0 atm, 85'K, 117 MPa.

Crazing Microparameters

For PCTFE, p was approximately equal to t, but for PMMA p was about l/lo t. It is interesting to note that for PCTFE, p was frequently greater than t and its crazes generally penetrated l/2 of the thickness of the specimen prior to fracture. The crazes in PMMA generally penetrated only about l/S the specimen thickness prior to fracture. Tables I and I1 for PCTFE and PMMA list values of p, 8, and p for the creep runs.

The same value of craze thickness a was obtained from both types of measurements and are given in Tables I and 11. Within a given specimen, a varied by about a factor of 2. The average values of a in Tables I and I1 are the averages of about 1% of the crazes in a given specimen compared to the large number of crazes that were observed at low magnification as in optical micrographs of Figure 2. Thus the uncertainty in the average value of a is much larger than the uncertainties in the average values of p, 8, and p . Although the uncertainties

CREEP EQUATION 1091

(b)

Fig. 3. Micrographs for determining p . Longitudinal sections that were metallographically polished. (a) PCTFE; (b) PMMA.

in the average values of p, l, and p are about f lo%, the uncertainty in a could be about f (25-50)%. Thus, the major source of random error in the calculation of the absolute value of the strain and the velocity comes from the uncertainty in the average value of a .

Retraction

The results of three retraction measurements are presented in Table 111. Specimens of PCTFE and PMMA were given a creep strain of a few percent. The permanent strain in a specimen was measured after it was warmed up to room temperature. It was found that most of the strain had been recovered in all cases. On the average 90% of the strain was recovered. It was of interest to know when the crazes retracted. After the specimen was unloaded at liquid- nitrogen temperature it was observed that only about 10% of the strain was recovered; when the unloaded specimen was warmed to room temperature, most


0 /

(b)

Fig. 4. Typical micrograph for measuring a , the thickness of the craze in PCTFE. (a) Replica in TEM; (b) SEM.

of the remaining strain was recovered. These observations were in agreement with those by Olf and Peterlin21 who observed that most of the craze strain which was produced in polypropylene at 78'K was recovered when the specimen was warmed up to room temperature and then unloaded.

These retractibility experiments show that q is about 0.09. According to eq. (12), b = 0 . 9 ~ for q = 0.09 and if p = 0.5.

COMPARISON OF THEORY AND EXPERIMENT

Strain

It is important to determine the validity of the basic equation, eq. (3). y, the

CREEP EQUATION 1093

I I I I I 10 12 14 16 18 20

C M I N.) (a)

lOATM.

.08 .75ATM,

.50 ATM.

TIME C M I N . 1

I-

0 5 10 15 20 2s TIME ( M I N I

(4 Fig. 5. Creep curves of PCTFE, Nz environment, under various conditions. (a) T = 78OK; P =

0.5 atm. (b) T = 78'K; u = 120.6 MPa. (c) P = 1.0 atm; (I = 117 MPa.

ratio of surface area to volume of the specimen, is given by

y = 2(2 + w ) / z w (13)

where w is the width of the specimen (5 mm) and z is the thickness. With b = 0.9a and a = 1, the strain according to theory is

€(theor) = 1.8pa&(z + w ) / z w (14)

The parameters for eq. (14) are given in Table I for 31 creep tests of PCTFE under a variety of conditions. In Figure 8, €(theor) is plotted against t(expt). The statistical correlation is 0.86. However, a least-squares plot gives a slope of 0.71 compared to a slope of 1.0 if €(theor) = 4expt). There is a systematic difference between theory and experiment. The only factor in the theory that was not based on a quantitative experimental observation was the assumption


0.4

0.3 I13

0.2

0. I

0 4 8 12 16 20

t ( rn in)

I I I I I I I , ,

0 6

0 5

04

0 3 p 3

0.2

01

0 4 8 12 16 20 t (m in )

0 4 8 12 16 20 t ( rn in)

Fig. 6. vs. t for PCTFE under various conditions. Short-time tests.

that the porosity of the crazes was 0.5. This assumption made b = 0.9a. In order for the slope in Figure 8 to be 1.0 requires b = 1.27a and a corresponding craze porosity of 0.59. Since no direct measurements of porosity in crazes formed in N2 at 78'K are available, it is suggested that a porosity of 0.6 for PCTFE is cur- rently the best estimate.

The theoretical strain was calculated for PMMA using the parameters in Table I1 and then plotted against c(expt) in Figure 9. The correlation coefficient is

CREEP EQUATION 1095

I I I I I I I I

0.3

0.2 EI/3

0. I

0

AT M

M Pa

I I I I I I I I

0 200 400 600 800 t ( min)

I I I I I

I I I 1

0 100 200 300 400 t (min)

Fig. 7. t1I3 vs. t for PCTFE under various conditions. Long-time tests.

0.99. The best fit between t(expt) and t(theor) corresponded to b = 0.7a, which gives a porosity of 0.4 for PMMA.

The high degree of correlation between t(theor) and c(expt) as shown in Figures 8 and 9 indicates that the theory is on the right track. Also if an average value of 0.5 for the porosity is used for both PCTFE and PMMA, then Figures 8 and 9 show that eq. (1) is capable of predicting the absolute value of the strain to within f30%.

Creep Equation Plots of vs. t (Figs. 6 and 7) are linear. The shape of the creep curves are

in complete agreement with the theoretical eq. (6). A more precise comparison with the theory is obtained by comparing the theoretical velocity u (theor) from eq. (9) with the experimental velocity u(expt), given by

u(expt) = AC/At

where AC is the change in length of craze during the time At and AC = C - t o , with CO being obtained from eq. (8). Using the values of s from Figures 6 and 7 and values of the microparameters from Table I, with a = 1 and b = 1.27a,


TABLE I Creep Tests of PCTFE in Nz

(atm) (crazes/mm2) (min) mm) (mm) (mm) (mm) (theor) (expt) P P t a e P z 6 t

1.00 0.79 0.75

1.00 0.75 0.50 0.25

1.00 0.75 0.50 0.35

0.85 0.75 0.50

1.00

1.00 0.75 0.50 0.25

1.00 0.75 0.50 0.25

0.50 0.25 0.15 0.10 0.05

0.25 0.15 0.05

85 66 62

185 113 85 19

151 132 85 21

34 28 15

77

272 219 139 64

360 250 132 66

398 378 303 96 0

886 648

20 100 510

4.2 6.8 18 817

5.2 12 48 322

55 111 540

105

2.7 6.8 14 445

1.5 4.1 5.4 49

1.6 3.6 9.3 38 318

1.4 2.8 8.4

u = 103 MPa, T = 78OK 0.56 0.90 0.52 0.46 0.87 0.55 0.72 0.95 0.51

u = 117 MPa, T = 78°K 0.87 0.50 0.60 0.44 0.49 0.63 0.50 0.47 0.64 1.10 0:72 0.63

u = 117 MPa, T = 80°K 0.76 0.43 0.49 0.55 0.45 0.65 0.24 0.53 0.62 0.26 0.58 0.60

u = 117 MPa, T = 85°K 0.78 0.56 0.63 0.63 0.40 0.57 0.34 0.84 0.45

u = 117 MPa, T = 88OK 0.33 1.3 0.65

u = 120.6 MPa, T = 78°K 0.47 0.36 0.72 0.51 0.46 0.63 0.87 0.40 0.75 0.52 0.39 0.72

u = 124 MPa, T = 78°K 0.52 0.40 0.60 0.37 0.41 0.66 0.56 0.51 0.65 0.35 0.40 0.59

u = 138 MPa, T = 78°K 0.23 0.31 0.55 0.36 0.19 0.58 0.33 0.31 0.61 0.48 0.43 0.60 0.00 0.00 0.00

u = 151.6 MPa, T = 78°K 0.27 0.19 0.58 0.44 0.18 0.56

0.953 0.998 0.886

1.031 0.876 0.866 0.925

1.021 1.039 1.041 0.894

1.062 1.031 0.826

0.894

1.026 0.998 1.034 1.019

0.894 1.062 0.912 0.996

0.978 0.963 0.864 0.953 0.00

0.927 1.049

0.050 0.031 0.052

0.10 0.037 0.031 0.022

0.051 0.045 0.014 0.0045

0.019 0.0084 0.0049

0.051

0.070 0.070 0.076 0.020

0.11 0.051 0.057 0.014

0.034 0.333 0.046 0.027 0.00

0.060 0.059

0.043 0.011 0.017

0.126 0.069 0.034 0.017

0.059 0.050 0.027 0.003

0.033 0.024 0.0022

0.054

0.104 0.091 0.059 0.032

0.114 0.074 0.050 0.011

0.048 0.030 0.023 0.024 0.001

0.045 0.029

227 .~ 0.32 0.19 0.58 1.026 0.017 0.011

u (theor) for PCTFE was calculated from the following equation

The results are plotted in Figure 10 and show excellent agreement between theory and experiment.

CREEP EQUATION 1097

TABLE I1 Creep Parameters of PMMA in Nz

P (crazes/ a E P Z t e Stress P T

Specimen mm2) (lO-3mm) (mm) (mm) (mm) (min) (expt) (MPa) (atm) ( O K )

A 25 0.80 0.97 0.079 0.676 12.2 0.0041 103.6 1.0 77.8 B 268 0.51 0.35 0.077 0.673 1.3 0.0070 110.2 1.0 77.8 C 11 0.57 1.51 0.104 0.681 3.94 0.0022 110.2 0.85 77.8 D 57 1.04 0.61 0.071 0.539 3.46 0.0099 117.3 0.75 77.8 E 82 0.54 0.46 0.090 0.785 0.58 0.0022 117.3 1.0 80.0

TABLE I11 Retraction Data

PCTFE(A) PCTFE(B) PMMA

Length before creep (in.) 0.86265 0.72573 0.95559 Length after retraction (in.) 0.86504 0.72872 0.95568

Total elongation during creep (in.) 0.0163 0.0320 0.0028

Permanent strain t total strain 0.15 0.09 0.03 q 0.12 0.08 0.03

Deformation after retraction (in.) 0.00239 0.00299 0.00009

Shear elongation during creep (in.) 0.00043 0.00047 0

a u = 117 MPa, P = 1.0 atm Nz; T = 78'K.

/ I

/ /

/ 1 l2 t t c

2 t- J. t- .

/ /

. .

.' 4 ' 1 ' b I 6 ' i I I b ' 1 1 2 ' I b ' I d l b

€ ( E X P I ( % I

Fig. 8. Theoretical versus experimental strain for PCTFE based on Table I. Dotted line is ideal curve for e(theor) = e(expt). e(theor) = p2(z + w ) 0.9 aEp/zw.

The intercept of a creep curve in Figures 6 and 7 gives the initial strain €0 in the specimen. The average value of the intercepts from all tests was 4I3(average) = 0.05 from Figures 6 and 7. This corresponds to an average initial strain of which is about equal to the experimental error in the strain measurement. Since all values of th'3 are positive, it appears that to is real. The calculation of an average value of l o from eq. (8) with the ~;/~(average) equal to 0.05, yields an &(average) of 0.09 mm. This value is the same magnitude (0.24 mm) which was


€ ( E X P I ( O h )

Fig. 9. Theoretical versus experimental strain for PMMA based on Table 11. Dotted line is c(theor) = t(expt). e(theor) = p2(2 + w ) 0.7 a-tpplzw.

I ' ' " " " I ' ' " " " 1 1

V ( e x p ) (rnrnlrnin)

Fig. 10. Theoretical versus experimental craze velocity for PCTFE based on Table I. Dotted line is u(theor) = u(expt).

obtained by Brown and Fischer'l from direct measurements of the growth velocity.

For PMMA, u(theor) was calculated using b = 0 . 7 ~ and the parameters in Table 11. For PMMA the total creep strain was small and the initial strain was negligible so u (theor) was calculated with the following modification of eq. (9):

u(theor) = - p -- t ( c c wz

where E is the creep strain and t is the time of the test. u(theor) vs. u(expt) is plotted in Figure 11; the agreement is excellent and strongly supports the validity of the theory.

CREEP EQUATION 1099

lo-2 lo+ 10-I I

V ( e x p ) (mm/min)

Fig. 11. Theoretical versus experimental craze velocity for PMMA based on Tables 11. Dotted line is u(theor) = u(expt).

DISCUSSION

The present observation that $13 varies linearly with t comes about if the craze grows proportionally in three dimensions and each dimension grows with a constant velocity. In general, it would be expected that would vary linearly with t where m = 1 , 2 , or 3 depending on the number of directions in which the craze grows and if it grows in each direction with a constant velocity. Noncon- stant growth velocities would lead to a different functional form for the creep equation.

Since the present theory assumes that dimensions of the craze maintain the same proportion to one another, the initial dimensions and the growth velocities are in the same proportion. If the theory were to be generalized by letting the initial dimensions of the craze have different proportionalities than those for the velocities, the shape of the creep curve would be altered very little in the range where the creep strain is much greater than €0.

The factors than determine the primordial thickness a0 are not known. a0 is important because it determines b. Since crazes originate at surface defects such as scratches, notches, voids, and dirt particles, it is suggested that a0 depends on the size and shape of the defect at which it nucleates. This suggestion is supported by the present investigation where it was observed that the thickness of crazes varied in each specimen and the average value of the thickness in a given specimen seemed to be independent of stress, temperature, or pressure of the gaseous crazing agent. It seems that the only variable that can determine vari- ations in craze thickness is a variability in the structure of the polymer. This variability in structure is most likely the spectrum of defects which can assist the stress in nucleating a craze.

Observations of the craze thickness (Fig. 4) indicated that the craze had a uniform thickness throughout most of its length except for a comparatively short region at the tip. These observations might lead to the false conclusion that the craze opening displacement b is also constant throughout the craze except at the region of the tip. However, the retraction experiments showed that there was almost complete retraction, and therefore, if b did increase continuously throughout the length of the craze, it might not be evident from observations of the retracted state.


CONCLUSIONS

1) The total strain in a craze specimen is given by the equation t = p y b A . 2) The creep curve from pure crazing has the form t1I3 = df3 + st, if the crazes

grow proportionally in three directions with a constant velocity. 3) The creep curve for pure crazing is completely determined by the density

of the crazes, their dimensions, and the growth velocity. 4) The craze opening displacement increases with time under stress and its

initial dimension probably depends on the size and shape of the defect at which the craze nucleated.

The major part of the support came from the Army Research Office. The central facilities supported by the NSF-MRL Program under grant No. DMR76-00678 were very helpful. Parts of the stipend for one of us (B.M.) came from a University Fellowship and from the NSF-MRL program.

References

1. N. Verheulpen-Heymans and J. C. Bauwens, J . Muter. Sci., 1 1 , l (1976). 2. N. Verheulpen-Heymans and J. C. Bauwens, J. Muter. Sci., 11,7 (1976). 3. N. Brown, Philos. Mug., 32,1041 (1975). 4. Y. Imai and N. Brown, Polymer, 18,298 (1977). 5. Y. Imai and N. Brown, J . Muter. Sci., 11,417 (1976). 6. N. Brown, Muter. Sci. Eng., 25,87 (1976). 7. W. G. Johnston and J. J. Gilman, J . Appl. Phys., 30,129 (1959). 8. J. Hoare and D. Hull, Philos. Mug., 26,443 (1972). 9. D. R. Curran, L. Seaman, and D. A. Shockey, Phys. Today, 30 ( l ) , 46 (1977).

10. N. Brown, J. Polym. Sci. Polym. Phys. Ed., 11,2099 (1973). 11. N. Brown and S. Fischer, J . Polym. Sci. Polym. Phys. Ed., 13,1315 (1975). 12. A. S. Argon, J. G. Hanoosh, and M. M. Salama, Fracture, 1, ICF 4,445 (1977). 13. S. Rabinowitz and P. Beardmore, CRC Crit. Reu. Mucromol. Sci., 1 , l (1972). 14. E. J. Kramer, H. G. Krenz, and D. G. Ast, J. Muter. Sci. (in press). 15. J. A. Sauer and C. C. Hsiao, Am. SOC. Mech. Eng. Trans., 75,895 (1953). 16. E. H. Andrews and G. M. Levy, Polymer, 15,599 (1974). 17. M. I. Bessonov and E. V. Kurshinskii, Fiz. Tuerd. Telu., 3,1314 (1961) [Sou. Phys. Solid State,

18. R. P. Kambour and A. S. Holik, J. Polym. Sci. A-2, 7,1393 (1969). 19. P. Beahan, M. Bevis, and D. Hull, Proc. R. SOC. Lond. Ser. A, 343,527 (1975). 20. E. L. Thomas and S. J. Israel, J. Muter. Sci., 10,1603 (1975). 21. H. G. Olf and A. Peterlin, J . Polym. Sci. Polym. Phys. Ed., 12,2209 (1974).

3,950 (1961)l.

Received March 23,1977 Revised December 13,1977

an equation for creep in terms of craze parameters

Documents