testing of theories of degradation dependent on the presence of weak links, e.g., tri or tetrafunctional branch points. Although the high values of z1 and x 2 , lvhich had to be a+ sumed in the present simulation and which are responsible for the clustering and the complexity of the branches, can be readily justified on steric grounds (Ehrlich and l\Iortimer, 1970), the low value n-hich must be assumed for A E , seems puzzling. It provides perhaps the major reason why the reaction mechanism used in these calculat’ions cannot be accepted without reservation.
Literature Cited
Ehrlich, P., J . Mucromol. Sci., Chem., A5, 1271 (1971). Ehrlich, p., Alortimer, G. A., Advun. Polym. Sci., 7, 386 (1970). Nicolas, L., J . Chim. Phys., 177 (1958). Roedel, hi,J , , J , A ~Chem. ~ sot., ~ 75,. 6110 (1953). Symcox, R. O., Ehrlich, P., ibid., 84, 531 (1962). Van der Molen, T. J., IUPAC Conference, Preprint 777, Bud&pest, Hungary, 1969. Weale, K. E., “Chemical Reaction a t High Pressures,” Spon, London, England, 1967. Willbourn, A. H., J . Polym. Sci., 34, 569 (1959). J. c,, Ehrlich, p., J . Chem, Sot,, 85, 1580 (1963).
Strain Softening and Yield of Polycarbonate-Moire-Grid Biaxial-Strain Analysis Morton H. Littl and Svenning Torp Division of JIacromolecular Science, Case Western Reserve University, Cleveland, OH 44106
When an external stress field is applied to a glassy polymer, it responds as follows. With increase in stress level, there is a gradual decrease in differential modulus (strain softening) until the glass fails (brittle failure), or i t becomes zero at which point the material is said to yield. Differential modulus is the slope of the stress-strain curve. h third, unrelated mode of failure is by crazing, which sometimes takes place in tension a t higher temperatures. Strain softening, yield, and crazing have received considerable attention in the literature. The effect of the type of stress field on the yield stress of a glassy polymer has been investigated by several workers. R h i t n e y and Andrew (1967) and Sternstein and Ongchin (1969) fitted their yield stress data to a Mohr and ColumbNavier-type pressure-sensitive, shear-stress yield criterion. The effect of normal stress on yield wis further investigated by Rabinoivitz et al. (1970). They ran experiments up to 7 K bar hydrostatic pressure and found t h a t shear yield stress had a linear dependence on pressure as predicted by the 1Iohr-type yield criteria. The effect of temperature and strain rate on the yield stress has been investigated by several workers. Holt (1968) found yield stress for P N M A to increase linearly with logarithmic strain rate over a n eight-decade range. Ekvall and Low (1964) and Robertson (1963) found that for PC the yield stress decreased almost linearly with increasing temperature, a t least away from the transitions. Lohr (1965a,b) also carried out experiments a t different strain rates and temperatures and then combined these results to construct yieldstress master curves by a method similar in concept to the construction of time-temperature master curves. The master curves obtained were linear, the slope being a material characteristic. The effect of temperature and stress level and of stress field on craze format ion was investigated by Sternstein and Ongchiii (1969) and Sternstein et al. (1968). They found t h a t the craze is generated by the normal stress component of the applied stress field and that the magnitude of this component required to generate a craze decreased linearly with increasing temperature. To whom correspondence should be addressed.
One common explanation for yield, first suggested by Bryant (1961) and later rephrased by Litt and Koch (1967) and Andrews and Kazama (1967), is to describe it as a stressinduced glass transition.
I n this context T ois not a temperature but a phenomenon dependent on temperature, stress, and time. With increase in temperature, stress, or time, T o will approach the test temperature, and the modulus of the glass will decrease. The glasses studied must be well characterized to determine the proper dependence of To on the above parameters. Isotropic materials undergoing reversible deformation can be fully characterized by two constants: shear modulus and bulk modulus or Young’s modulus and Poisson’s ratio. We will discuss the latter two. Most of the literature today contains uniaxial tension data from which Young’s modulus can be obtained, but little has been done about Poisson’s ratio. Nielseii (1965) determined Poisson’s ratio for some glassy polymers by a stepmise load-increment tensional test. His values are ambiguous a t higher strains owing to the method and difficult to analyze as the strain rate is not constant. Whitney and Andrews (1967) have also studied some glassy polymers by a uniaxial compression dilatometer. However, there is still considerable ambiguity about Poisson’s ratio and its dependence on strain, temperature, and strain rate. With only the knowledge of Young’s modulus, the material is only partly characterized. A detailed knowledge of Poisson’s ratio should give additional information about the yielding process. The change in Poisson’s ratio, or volume, with stress in a uniaxial stress test seemed to be the simplest obtained for a n isotropic material by axial and transverse strain measurements. There are several strain-measurement devices available on the market today, and we decided to employ a newly developed one-by moire interference patterns. This method was chosen because it yields biaxial strain to a high degree of accuracy in the desired strain range, experimental technique Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
357
lncldent Llght
AV/V = ea’(l
SDeClme”
Figure 1 . Moird interference pattern generation from two adjacent grids (adapted from literature of Photolastic, Inc., Malvern, PA)
and data analyses are simple, and sample preparation is straightforward. Theory
Stress. Any stress field can be divided into normal and shear stress. For a n elastic material normal stress will dilate or compress the material, depending upon the loading, whereas shear stress will change the shape of t h e specimen with no effect on volume. Some basic definitions are listed: For normal stress or hydrostatic pressure: Hydrostatic pressure, p Volume, v Bulk modulus, K = - p / A v / v
+
Djd’ = D / d - 1 or D
E
= 7/y
€1
E
Maximum shear stress, 7,118x = u / 2 Average hydrostatic pressure, p = u/3 The three moduli defined above can be interrelated with the aid of Poisson’s ratio by the following standard mechanics expression :
+
Y)
=
3 K(l
- 2 V)
(1)
For glassy polymers there are further complications. -411 the moduli are dependent on the strain level and the strain rate. Consequently, the above relationships must be used in differential form, and strain rates must always be quoted. Poisson’s ratio as obtained from the axial and tangential strain measurements can be converted to volume change by the following relationship, ignoring second and higher order terms, 358
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
-
D
=
1 1-
- 1
~
€E
and true strain, e, by:
The uniaxial tensile or compressive stress field is a mixture of shear and hydrostatic loading. I n this field:
2 G(l
- _d
d’-d = y d
E
Eulerian strain can thus be obtained simply by dividing the pitch of the master by the pitch of the generated fringes. From Eulerian strain, engineering strain, e‘, can be obtained by the following expression:
Tensile stress, u Axial and tangential true and engineering strains, respectively, ea, E t , ea ’ , e t I Poisson’s ratio, Y = e t / e a Young’s modulus, E = u/ea
=
dd’ d‘ - d
= -
Eulerian strain,
For uniaxial tensile or compressive stress:
E
(2)
Moir6 Method for Strain Analysis. The method can be best explained by referring t o Figure 1. For t h e sake of simplicity, consider the transparent specimen t o have a set of parallel black lines applied to its surface. Directly on top of t’he sample is placed a master m-ith a n identical set of lines. With t h e sample undeformed t h e densitmyof lines is the same for t’he master and t h e specimen. T h e transmitted light will be of uniform intensity; if t h e lines coincide, the intensity will be maximum, and if t,he lines fall in between, it will be minimum. When the sample is s h i n e d , the lines will deform with it so that the line density changes. Thus, in some areas the lines of the two parts will coincide to give maximum transmission, whereas in other areas the lines will fall in betveen each other, giving minimum transmission. Strain Xeasurement. The actual strain can be calculated from knowledge of density of the master and density of the interference fringes (Holister, 1967). Assume both grids initially t o have a pitch, d. Then with a strain, E, applied perpendicular to t’he lines, the pitch of the specimen grid is changed to d’. If the distance between adjacent fringes developed is D , then the number of grid lines in this distance is D / d for the master and Djd’ for the sample. For tensile strain Did’ = D j d - 1,since the distance from one grid to the next contains one less line than the undeformed master. By a similar argument, for compression D/d’ = D j d 1. Hence, for tension
For shear stress: Shear stress, 7 Shear strain, y Shear modulus, G
- 2 v)
=
In ( I
+
E’)
If biaxial measurement is desired, as in this work, grids with an array of lines in two perpendicular directions are used; however, the principles and analyses are the same as for the parallel line case. I n a uniaxial test d h grid lines perpendicular to the axial and transverse directions, the fringes developed should also be perpendicular to these directions, and the magnitudes of the principal strains are simply obtained from the fringe density. =iny strain field can be analyzed by the biaxial moiri. method, as described by Holister (1967). Apparent Strains. There are several experimental inaccuracies that would tend to generate fringes without any strain, termed apparent strain, eapp. The most important causes of these difficulties are discussed. Angular mismatch of grids: If the two grids are not aligned, fringes will be generated without strain. The direction arid frequency of the fringes will depend on the magnitude of the mismatch and the applied strain (Holister, 1967). If the surfaces to which the two grids are bonded are not parallel,
fringes will be generated without strain. A t zero strain the pitch, DaPp,of fringes owing to mismatch CY of grid planes will be: d
Dapp =
Or Eapp cos ff
= COS
n
II
CY
This apparent strain will be independent of strain level and can thus simply be subtracted. Experimental
Materials Used. Polymer. All t h e work h a s been done on Bisphenol-A Polycarbonate (PC), manufactured by General Electric Co. as uv-stabilized Lexan sheet. P C was chosen for several reasons: It is a commercial glassy polymer; i t has received considerable attention by several workers and is therefore well characterized. T , is a t about 145OC, whereas the 7-transition is around - 100°C (Matsuoka and Ishida, 1966). Throughout most of this range the material fails by ductile yield. Hence, over a temperature range of more than 2OO0C, the molecular motion is compatible, and the failure is by yield. It is, therefore, a highly suitable material for a study of the effect of temperature on Poisson's ratio and Young's modulus. Grids. The grids used were supplied by Photolastic, Inc., Malvern, PA. The master grid consisted of a perpendicular array of metal lines with a density of 1000 lines/in. in both z and y directions, deposited on flawless soda-lime glass supplied by Eastman Kodak (VPAI # 1-1000). The sample grid consisted of a n array of square metal dots in the z and y directions with a pitch similar to the master, deposited on a thin stainless-steel backing plate (VI'S # 16-1000). This grid was applied to the sample by a technique called "Transferable Grid Xethod," developed by F. Zandmann and D. Post of Vishay Instruments, Inc. The grid is cemented to the specimen by a low-modulus, high-elongation, transparent epoxy adhesive (VPAC and VPH-1). After the resin is cured, the backing plate is stripped off, leaving the high accuracy pattern in place with no initial distortion. Since a low-modulus adhesive is used, the reinforcement is negligible. As the sample grid consists of noninterconnected dots, the regular array will not break down even a t high elongation. Experimental Technique. Preparation of Samples. A 5.5 X 8.5-sq in. sheet was c u t from 60-mil thick commercial Lexan plate. The sheet was dried a t 140°C in a vacuum oven overnight. The dried sheet was clamped between two polished 1/16-in. thick aluminum plates and heated in a n air oven a t 250°C, just above the melting point, for 15-20 min. It was then quenched in ice water. Dog bone-shaped specimens were cut from the sheet by a sidemill Chapman Photoelastic Model Maker to dimensions shown in Figure 2 , by use of a template guide. Altogether, 65 specimens were cut and annealed in a silicone-oil temperature bath as follows: 18 hr a t 143.3OC, 51 h r a t 141.3"C, 15 h r a t 140.3"C, 1 h r a t 139.3'C, and then slowlycooled to 129OC. For the application of the grids, the sample surface was carefully cleaned by methanol and distilled water. The bondable grids, cut by a paper cutter to 0.5-in. squares, were also cleaned by methanol and placed in the middle of the specimen. The area around the grid was masked by tape to keep the grid in place and exclude unwanted cement. The grid was removed, adhesive applied to the specimen, and the grid carefully placed 011 top. The best cementing results were achieved by placing a 130-gram weight on top of t h e grid and by curing in a n air oven a t 60°C. After curing, the backing plate and masking tape were stripped off.
Figure 2. Sample dimensions
Figure 3. Apparatus for moiri-method strain analysis
Stress and Strain Measurements. The apparatus built is shown in Figure 3. It had two basic functions: to hold the specimen in the Instron testing machine and to hold the master grid next t o the sample grid. The jaws were machined from stainless steel aligned with the sample by placing them in a channel of the same breadth a s the specimen and the jaws. For better alignment, spacers were put between the middle section of the sample and the channel wall. The master grid was supported by a n aluminum carriage t h a t facilitated accurate angular positioning by a 6.4:1 gear drive for proper alignment. The carriage was held in place by four sliding points screwed to the jaws. Each of these could be adjusted to put the two grid planes parallel and close to each other. The master grid was adjusted after the assembly was p u t in the Instron and lightly loaded. The four sliding points were adjusted until the grids were parallel and just in contact; then the master was rotated until the minimum number of, or no fringes occurred. Temperature Control. The temperature controller is shown in Figure 4. It was designed t o give only small (*0.5"C) temperature fluctuations from - 100°C to S14O"C over long periods of time and no thermal gradient in the sample region. ,4Dewar flask, manufactured by H. S. Martin & Son, was used for the testing chamber. As shown in Figure 4, i t has a 1.5-in. diam optical flat window set in a 1.5-in. wide unsilvered horizontal strip. The window allowed a n undistorted view of the interference fringes, and the grids were illuminated through the unsilvered portion. Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
359
T
E
3 I
f $4
Figure 4. Heat chamber
VALVE
Figure 5. Cold constant-temperature air controller The jaws were supported in the Dewar flask by the compression member also shown in Figure 4. It consisted of a n aluminum cage of two 0.5-in. thick plates held apart by four 5/8-in. diam tubes. The constant temperature air entering the heat chamber was first passed through these tubes t o average out any temperature fluctuations. The cage was supported by a n 8-in. long, thin-walled stainless-steel tube. A 3/8-in. diam stainless-steel rod served as the tension member to transmit the load from the sample t o the load cell. Constant temperature air was fed into the heat chamber. Compressed air from the main was dried by passing it through molecular sieves. Below room temperature: The air was passed through a coil in a Dewar flask containing a solvent with a desired freezing point a s shown in Figure 5. The bath was kept a t t h a t temperature by having some liquid and solid present. This was achieved by passing liquid nitrogen through a second coil on which the solid deposited. The flow of liquid nitrogen was kept relatively constant by keeping a constant pressure over the cryogenic bottle. The temperature levels were changed by using solvents with different freezing points. Above room temperature: I n this case the air was heated by passing i t through a coil in a constant-temperature silicone oil bath. Fine temperature adjustments were achieved by changing the aidlow. The experiments were carried out after the system had reached thermal equilibrium. Aided by changing the air flow and passing liquid nitrogen through the apparatus a t low temperature, this equilibrium was usually reached in 1-2 hr. The temperature was measured by a thermocouple held close to the sample. 360
Ind. Ens. Chem. Prod. Res. Develop.. Vol. 11. No. 3, 1972
Figure 6. Typical sequence of moirC! fringe formation with strain (Eulerian strains) Experimental Procedure. T h e whole assembly was set u p as described in an Instron testing machine, Model T T C M L . A sodium vapor lamp was used for a light source as the monochromatic light seemed t o yield more distinct interference patterns. Pictures of the moire patterns were taken with a 35-mm single-lens reflex camera, by use of lens spacers t o get sufficiently close t o the subject. T h e camera was loaded with Kodak high contrast copy film, and the pictures were shot a t 1/125-sec exposure time. This was sufficiently rapid t o “freeze” the moving moire fringes. Pictures were taken roughly every 0.35% strain, and a mark was simultaneously put on the Instron chart paper to get the load a t that particular strain. The pictures were developed and magnified to about five times the actual pattern size; extracts from a typical sequence for stress-strain are shown in Figure 6. The pitch of the grids was measured in the axial and transverse directions with a ruler. From this Eulerian, engineering and true axial and transverse strains and engineering stresses were calculated. Results and Discussion Stress and Strain Measurements. T h e axial and transverse strains were calculated from the pictures taken of the interference fringes, and the tensile loads were recorded simultaneously with each picture and converted t o engineering stresses. The error in the strain determination depends on the accuracies of the fringe density measurement and of grid alignment. I n the 0&5% strain range with 1000 lines/in. grids, the error should he less than 10.01% strain. The load was read from the Instron chart paper t o ahout f l kg; with the sample dimensions used, this represents about 1 5 kg/cm2 error in stress. This accuracy was, however, reduced by a slight nonuniformity of specimen cross section. For the crosssectional area and the fringe pitch measurement, the average in the moir&grid region was used. It was estimated t h a t the resulting accuracy was 1 2 % in stress measurement, whereas strain was measured to f0.03% absolute. The engineering stresses and axial engineering strains are plotted in Figure 7 for different temperature levels a t the same strain rate and in Figure 8 for different strain rates a t room temperature, The behavior is as expected with an increase in Young’s modulus and yield stress with a decrease in temperature or increase in strain rate. A t least two runs were made a t any temperature and strain rate.
TEMP: t26
OC
STRAIN RATES IN *rlmln
d " - I!
: 2 3 AXIAL TRUE STRAIN
0
1
5 9 0
Figure 10. True axial strain as function of true transverse strain for different strain rates
TEHPERATVRES
Figure 7. Engineering stress as function of axial engineering strain for different temperatures
T
0:
-65
mi
-25.C
.:.:+ v:
26s
+65T +88'C
ENGR. STRAIN,
:E
Figure 1 1. Tangential Young's modulus as function of axial engineering strain for different temperatures
1
Et
A STRAIN RATES
0
0 ENGR. STPAIN,
6,
0 :
7.8
gdmin
o:
.76
?afmin
T:
978 9dmin a0378 %/rnin
e:
Figure 8. Engineering stress as function of axial engineering strain for different strain rates
'. .-..
..
O C 0
I
2 3 ,4 ENGR. ST PAIN,€^
1
570
Figure 12. Tangential Young's modulus as function of axial engineering strain for different strain rates
The strains measured by the moire method were converted
to true strains, and the axial strain was plotted vs. the tan-
3
I 2 3 A X I A L TRUE S T R A I N
4
5
6
7 %
Figure 9. True axial strain as function of true transverse strain for different temperatures
gential strain in Figure 9 for different temperatures and in Figure 10 for different strain rates. The different initial strains or prestrains shown for the isotherms in Figure 9 are due to the difference in temperature a t which the grid was cemented to the specimen and experiment conducted. Tangential Modulus. From Figures 7 and 8, the tangential modulus was calculated from the slope a t every 0,5y0strain increment and plotted vs. axial engineering strain as shown in Figures 11 and 12. The behavior is as Ind. Eng. Chem. Prod. Res. Develop., Vol. 11, No. 3, 1972
361
DIFFERENTIAL WISSON’S RATIO
DIFFERENTIAL POISSON’S RATIO
STRAIN RATE: 8 7 a & 9
TEMR:+26
OC
‘E I
T
C26 oc
+ 0
0
6:
o..o
84
I
,
,
,
0
1
2
3
ENOR. ST&
‘
+
%
I
Figure 13. Calculated (-) and measured ( 0 ) Poisson’s ratio as function af axial engineering strain a t different temperatures
expected, showing a n increase in initial tangential modulus with decrease in temperature or increase in strain rate, and a monotonic decrease in tangential modulus with straining. From Figure 11, the shape of the curves vary with temperature. At the low temperatures the decay is almost linear with strain, whereas a t higher temperatures the decay is initially smaller and then increases. At the two highest temperatures, 65” and 88’C, the curves show a discontinuity a t yield. Observations of specimens drawn a t the two higher temperatures reveal pronounced whitening owing to shear bands in the neck region and crazes throughout the sample. Also, the angle of the neck increases from the maximum shear position taken by the lower temperature specimens to about 78’ a t 65°C and about 88’ at 88°C. This would indicate that failure is caused by the normal stress, generating crazes t h a t weaken the material to give premature yielding. For all the tangential modulus data to be compatible, craze formation was assumed only to occur close to yield for these two hightemperature samples. Further microscopic studies showed t h a t no crazes developed at the two lowest temperatures although there were some present close to the neck in the room-temperature sample. From Figure 12, for three decades of strain rate, the central portions of the curves coincided, whereas the lowest strain-rate behavior was distinctly different. Observations of the lowest strain-rate specimen indicate the same features of vertical neck, shear-band whitening, and crazing as described for the high-temperature specimens. Part of the tangential modulus softening of the low-rate specimen could thus probably be attributed to craze formation, a t least a t higher strains. Poisson’s Ratio. As stated in t h e theory section, Poisson’s ratio is t h e negative ratio of transverse and axial true strains; for a glassy polymer this ratio must be used in differential form. This is equivalent t o the slope of t h e curve of transverse true strain plotted vs. axial true strain as given in Figures 9 and 10. From these curves, Poisson’s ratio was calculated from t h e slope at every 0.5y0 axial strain interval and plotted vs. axial strain as shown by t h e experimental points in Figures 13 and 14. 362
~ 7 8wmin
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
I
0
2 3 ENGR. STRAIN
4
e:
5 %
Figure 14. Calculated (-) and measured ( 0 ) Poisson’s ratio as function of axial engineering strain at different strain rates
I n the Theory section a n expression was given relating Young’s and bulk modulus with Poisson’s ratio for a n elastic material. I n this work tangential Young’s modulus was determined experimentally (Figures l l and la), so that by the additional knowledge of bulk modulus, a n “ideal” Poisson’s ratio can be calculated. Bulk modulus for P C was obtained indirectly from the work of the following researchers: Natsuoka and Ishida (1966) determined specific volume vs. temperature a t various pressure levels. Their data were replotted to give the following relationships of bulk modulus to temperature T,
K
=
[4.0
x
lo4 - 25(T
- 25)1kg/cm2
Warfield (1967) gave a plot of pressure vs. change in volume a t room temperature. The slope of this curve gives bulk modulus as a function of pressure, P , in kg/cm2
K
=
[4.2 X lo4
+ 13 P]kg/cm2
Hennig (1965) estimated linear compressibility for amorphous P C a t room temperature to be 7.92 X I O-6cm2/kg. This can be converted to bulk modulus,
K=-
3
x
7.92
x
=
4.20 X lo4kg/cm2
The literature values obtained agree. The values obtained from Matsuoka and Ishida were used to get bulk modulus as a function of temperature. The effect of pressure was estimated from ffarfield’s data. In tension the reduction in bulk modulus owing to normal dilational stress seemed to stay below 10%. This is small compared with the change in tangential Young’s modulus; hence, bulk modulus was assumed to be constant with stress. The “ideal” values of Poisson’s ratio thus calculated from Figure 1 are shown by the solid curves in Figures 13 and 14. The “ideal” Poisson’s ratio represents the volume change owing to the normal stress component, whereas the one experimentally determined shows the actual volume change. Thus, the difference between the two should represent the possible volume effects of relaxation and/or shear stress that could occur in a glassy polymer. High-accuracy determina-
tion of Poisson’s ratio is difficult, and the values obtained in this series of esperiments are only sufficiently accurate for a qualitative discussion. -It lower strains the “ideal” and csperimental values for Poisson’s ratio agree reasonably well, thus indicating that the mat’erial responded as if elastic in this region. - i t higher strains the “ideal” Poisson’s ratio approaches 0.5 as espected, as the tangential Young’s modulus tends t o zero. From the espression given in the Theory section relating 1-olume change to Poisson’s ratio, this is the value a t n-hich no volume change takes place. However, for several runs the experimental Poisson’s ratio esceeds 0.5 close to j-ield! This indicates t h a t near yield there is a decrease in voliuiie n-hile stress is increasing. Close to yield, the material contracts iii spite of the dilatoric normal stress. This phenomenon lean be explained in terms of relasation of the imposed hydrostatic strain. At higher strains the molecules are sufficiently activated to relax, and their rearraiigemelit could result in a densification by relasing out some of tlie volume increase introduced during the straining process, facilitating better packing of the chains. Summary and Conclusions
Method. Iu a uniasial stress test on a n isotropic material, volume changes with strain could be calculated by biaxial strain measurements. The moir6 met’hod was highly suitable for biaxial strain analysis of glassy polymers. I t yielded simultaneous strain measurements of high accuracy a t any desired strain rate, simply b y photographing t’he interference patterns. Furthermore, t h e sample preparation was simple and did not interfere with t h e mechanical properties of t.he sample. Froni t h e iiiterfereiice patterns t h e strain can be “seen” directly, and the strain system resulting from the applied stress can be directly checked. Ailso,the method provided a direct strain me:wirement in a desired portion of the sample, and thus, “end effects,” which occur with overall elongation measureinelits, were automatically eliminated. Tangential Modules. T h e initial tangential Young’s modulus of t h e glassy polymer increased, as expected, with decrease in temperature and increase in strain rate. K i t h straining, t h e modulus decreased continuously until it was zero, at, which point t h e material was said t o yield. There is no linear elastic region in annealed polycarbonate when stress-strain esperiments are t h e criteria. This is contrary t’o t h e assumptions of many investigators, b u t t h e d a t a are accurate enough to lend confideiice in (,he above conclusions. Poisson’s Ratio. The iiiitial values of “ideal” Poisson’s ratio decreased with decrease in temperature and increase in strain rate and varied from 0.38 t o 0.41. K i t h increase in strain, Poisson’s ratio steadily increased until it reached 0.5 a t yield. T h e values of Po;sson’s ratio obtained from the biaxial-strain analysis shelved good agreement a t lower strains. However, a t larger strains a discrepancy appeared. and around 5% strain the measured values indicated :i densification with further straining. This was attributed to partial relaxation of the dilation caused by the straining process.
Three-dimensional relaxation before yield during drawing helped esplain several phenomena in tlie literature. Litt and Koch (1967) found t h a t specimens of BPA polycarbonate biaxially rolled to less than 50% of their original thickness were thicker after drawing than before. This was postulated and is now known to be owing to relaxation of the compression when strain is such that large scale relaxations in three dimensions occur. Vincent (1960) noticed that samples stressed to near or past yield and then returned to zero stress rapidly were restressed along the same curve with 110 yield masimuni. When lie held the sample a t constant strain for some time, then returned t o zero and restrained, a yield peak was found as in the original uristrained specimen. This can be rationalized by volume relasation on holding a t constant strain. This creates a iierv structure in the neck which is iio longer in equilibrium with the stress. Greater stress must then be applied to bring it to yield. Rapid destraining prevents relasation, and in restraining the initial structure is obtained which yields a t the original yield stress. Strain Softening and Yield. Yield is the ultimate result of strain softening. At lower strains t,he Poisson’s rat,io d a t a compare and therefore indicate a reversible behavior of t h e material. T h e st,raiii softening in this region is also believed to be mainly reversible, Le., mainly elastic. At higher strains the Poisson’s ratio data, lion-ever, do not compare. The strain softening in this region is therefore believed to be mainly plastic. At present there are not sufficient data arailable t o separate the two strain-softening modes arid thus evaluate their importance. From the data in the literature, strain softening and yield are primarily dependent upon the shear component of the applied stress. The normal stress component n-ill change the levels of shear stress required for the material to yield, probably in a manner similar to that of temperature. complete description of yield must include the effect of temperature, pressure, time, and shear-stress level. literature Cited
Andrea-s, R . I)., Kazama, Y., J . A p p l . Phys., 38, 4118 (1967). Bryant, G. Jl., Text. Res. J . , 31, 399 11961). Ekvall, R . h.,Low, Jr., J. Ii., J . A p p l . Polym. Sei., 8, 1677 11464) ,--~-,. Hennig, J., Kolloid Z., 202? 127 (1966). Holister, G. S., “Experimental Stress Analysis,” Cambridge Univ. Press, Cambridge, England, 1967. Holt, S. L., J . A p p l . Polym. Sei., 12, 1653 (1968). Litt, ll.,Koch, P., Polym. Lett., 5 , 251 (1967). Lohr, J. J., A p p l . Polym. S y m p . , 1, 5.5 (1965a). Lohr, J. J., Trans. Soc. Rheol., 9 ( l ) ,65 (1965b). JIatsuoka, S., Ishida, Y., J . P o l y n . Sci., 14, 247 (1966). Xielsen, L. E., Trans. Soc. Xheol., 9 ( l ) ,243 (1965). Rabinowitz, S.,Ward, I. AI., Parry, J. S. C., J . Jlafer. Sei., 5 , 29 (1970). Robertson, R. E., J . A p p l . Polyni. Sei., 7, 443 (1963). Sternstein, S. S., Ongchin, L., Polyni. Prepr., 10 (a),1117 (September 1969). Sternstein, S.S., Ongchin, L., Silverman, A, A p p l . Polurn. S y n i p . , 7, 175 (1968). Vincent, P.I., Polymer, 1, 7 (1960). Warfield, R . W., J . A p p l . Chem., 17, 263 (1967). Whitnev. W..Andrevis. R . 11.. J . Polurn. Sei.. Part C. 16, 2981 (1965): Abstracted from the LIS thesis of S.Torp (1970).
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
363
