On the phenomenology of yield in bisphenol-A polycarbonate.

INTRODUCTION

The results in this paper are part of a program aimed at developing thermomechanical constitutive equations and yield criteria for engineering thermoplastics. Much of the work on characterizing the mechanical behavior of such materials - test types, interpretation of test data, constitutive equations, and yield criteria - and the use of mechanical data for predicting part performance are based on a metals mindset. This approach would be appropriate if the mechanical behavior of plastics were qualitatively similar to that of metals, the only difference being in the magnitudes of the various parameters. But plastics are known to exhibit phenomena that metals do not: Plastics undergo stable necking (1-5), they can craze (6, 7), and their yield behavior is significantly affected by hydrostatic pressure (8-11). Also, they can fully recover from a yielded state on being heated to the glass transition temperature. Clearly, then, constitutive modeling must be based on an understanding of the different response phenomena that occur when such materials are subjected to thermomechanical loads. A general review of the yield (12) and post-yield (13) behavior of glassy polymers, and other aspects of mechanical behavior, such as creep, are addressed in Haward (14). This paper explores the phenomenology of yield in bisphenol-A polycarbonate (PC) through several different tests. All the specimens were cut from extruded sheets of a commercially available grade (GE Plastics' Lexan 9030) of PC.

The tensile test is the most commonly used method for determining the mechanical properties of materials. However, because polymers yield in a tensile test by deformation localization, conducting such a test at a constant deformation rate is difficult and generating true stress-strain curves (5) is a difficult task. Hourglass specimens have been used to ensure that yield initiates in a specified region where the strains can then be measured (15-17). Techniques have been developed to use strain feedback to control the strain rate in such specimens (15, 18-20), making it possible to obtain true stress-strain curves. Buisson and Ravi-Chandar (21) have investigated the true-stress, true-strain relationship of PC in uniaxial tension. The strain field was measured by monitoring the deformation of an imprinted grid; the stress field was determined by using a photoelastic technique. However, there is some question as to the one-dimensionality of the data obtained through all tensile tests.

Other types of tests can be used for mapping mechanical properties. A representative list follows: Titomanlio and Rizzo (22) studied the creep and relaxation behavior of PC in compression. Arruda et al. (23) showed that the stress-strain curves in uniaxial compression and plane strain compression are different. Wu and Turner (24) studied the yield behavior of PC through torsion tests on thin-walled tubes. G'Sell et al. (25) used a plane simple shear test for studying the deformation of solid polymers at large strains. Spitzig and Richmond (26) studied the effect of hydrostatic pressure on the deformation of polyethylene and PC in tension and compression. Caddell and Kim (27) considered the influence of hydrostatic pressure on the yield of PC. Yee and Carapellucci (28) mapped the biaxial deformation and yield behavior of PC. Other papers concerned with yielding include Stern-stein et al. (29) Sternstein and Ongchin (30), Stern-stein and Meyers (31), and Yee and Detorres (32).

The current state of understanding of yield in narrow, rectangular cross-sectioned tensile specimens can be explained as follows. When a specimen is pulled at a constant displacement rate, the load first increases, attains a maximum, and then drops off precipitously to a lower value, at which point a neck is formed. Further extension results in more of the unnecked material undergoing necking, i.e., in stable neck propagation at a constant load, during which the strain in the necked material remains constant. When the neck reaches the shoulders of the specimen, the load begins to increase, resulting in an increase in the strain in the neck. Since the load depends on the cross-sectional area, it is more appropriate to use the nominal stress [[Sigma].sub.n], the load divided by the original cross-sectional area, to describe the phenomenon. The solid curve in Fig. 1 shows the variation of the nominal stress versus the displacement for PC. The specimen stretches homogeneously from 0 to A, where the stress attains a maximum, critical value [[Sigma].sub.0] = [[Sigma].sub.A]. At this point the stress drops off very rapidly to the draw stress [[Sigma].sub.d] = [[Sigma].sub.B]. The nominal stress starts to increase only after the necked material reaches the shoulders of the specimen. For comparison, corresponding stress-displacement curves are also shown by dashed lines for the high-temperature, amorphous polymer polyetherimide (PEI) and for the semicrystalline polymer poly(butylene terephthalate) (PBT). The responses of the two amorphous polymers, PC and PEI, are similar; however, the response of the semicrystalline PBT is significantly different (note the portion A[prime]B[prime]C[prime]D[prime]). Figure 2 shows the approximate true stress-stretch curve for PC, corresponding to the load-displacement curve in Fig. 1. The stress and stretch increase homogeneously till point A, where the stretch is about [Lambda] = 1.06. At this point the specimen necks at some location; the stretch in the necked material is about 1.7. Because the shift from state A to B occurs almost instantaneously, the material undergoes an "instantaneous" jump in stretch from [Lambda] = 1.06 to [Lambda] = 1.7, resulting in a second homogeneous deformation field. Thus, the one-dimensional stress-stretch curve is only determined between 0 and A and between B and C. What happens between A and B has not been characterized - hence the dashed line between A and B in Fig. 2. This experiment raises several interesting questions: First, what happens when the specimen is subjected to a constant stress between the draw stress [[Sigma].sub.d] and the threshold, or critical, stress [[Sigma].sub.O]? Second, what occurs between points A and B during which the material yields to form a neck? Third, how are [[Sigma].sub.0] and [[Sigma].sub.d] affected by deformation rate and temperature? And fourth, what is the phenomenology of yield for a multiaxial deformation field? The temperatures at which thermally Induced recovery from yielded states occur are also of interest.

This paper explores answers to these questions through two types of experiments: (1) tensile tests in which the widths of the specimens were varied over a wide range with the aim of observing differences between plane-stress and plane-strain extension, and (ii) bulge tests, in which circular sheets were subjected to biaxial deformation fields through lateral pressure-induced bulging.

EFFECT OF DEFORMATION RATE AND TEMPERATURE ON THE LOAD-STRETCH BEHAVIOR IN A TENSILE TEST

This section considers how the critical stress [[Sigma].sub.0] and the draw stress [[Sigma].sub.d] are affected by the strain rate and temperature. By attaching extensometers to a specimen to monitor the axial and transverse stretches during a tensile test, the load-displacement behavior in Fig. 1 can be monitored in the form of a true-stress versus stretch curve. However, in the preliminary experiments described in this section, the stretch was only monitored in the axial direction. Therefore, the results will be presented in the form of stress-stretch plots, in which stress will be the nominal stress [[Sigma].sub.n] based on the original area of the specimen and is really a measure for the load normalized by the specimen cross-sectional area.

Standard 2.3-mm-(0.09-in-) thick ASTM D638 specimens were pulled in tension at constant displacement rates. corresponding to nominal stretch (strain) rates of [Mathematical Expression Omitted] and [10.sup.-3] [10.sup.-2] [10.sup.-1] [10.sup.0] [s.sup.-1] at temperatures of 22, 37.5, 51.5, and 65.5 [degrees] C (72, 100, 125, and 150 [degrees] F). In these tests, the stress-stretch, or stress-strain, curve has the typical shape shown in Fig. 3. The nominal stress increases up to the point A (critical stress [[Sigma].sub.0]) at which, on necking, the stress suddenly drops to the draw stress [[Sigma].sub.d] [ILLUSTRATION FOR FIGURE 1 OMITTED]. On neck initiation, the stretch in the necked material Increases from 1.06 to 1.7. However, in the unnecked material, the stretch (strain) actually decreases instantaneously, from A to [B.sub.1]. For convenience, the portion A [B.sub.1] of the stress-stretch curve will be simplified to A [B.sub.2], as indicated in Fig. 3.

The stress-stretch curves at the five strain rates at 22 [degrees] C (72 [degrees] F) are shown in Fig. 4, and the corresponding curves at 65.5 [degrees] C (150 [degrees] F) are shown in Fig. 5. While the magnitudes of [[Sigma].sub.0] and [[Sigma].sub.d] are affected by both the strain rate and the temperature - [[Sigma].sub.0] and [[Sigma].sub.d] increase with increasing strain rates but decrease with increasing temperatures - the ratio [[Sigma].sub.d]/[[Sigma].sub.0] appears to be insensitive to these two parameters, as can be seen from Fig. 6. This figure also shows that [[Sigma].sub.d]/[[Sigma].sub.0] has a nominal value of about 0.75 for strain rates between [10.sup.-4] to [10.sup.0] [s.sup.-1] and temperatures between 22 and 65.5 [degrees] C.

It may appear from Fig. 4 that, at room temperature, the stretch at which the load drops off suddenly increases from about 1.06 to 1.07 as the strain rate increase from [10.sup.-4] to [10.sup.0] [s.sup.-1], and from Fig. 5 that this change in the stretch is smaller at the higher temperature of 65.5 [degrees] C. However, these tests were not sensitive enough to quantify the effect of strain rate on the stretch at which the load falls off.

The actual (true) stress [Sigma] resulting from a tensile load will be larger than the nominal stress [[Sigma].sub.n] based on the original cross-sectional area of the specimen. Estimates for [Sigma] can be obtained by assigning a suitable Poisson's ratio. Consider a thin rectangular specimen of initial width [b.sub.0] and thickness [t.sub.0] that is stretched in the longitudinal direction to a stretch [[Lambda].sub.1], at which the width and the thickness are b and t, respectively. Let the stretches in the width and thickness directions be [[Lambda].sub.2] and [[Lambda].sub.3], respectively. Then, the initial ([A.sub.0]) and final (A) cross-sectional areas of the specimen, and the initial ([V.sub.0]) and final (V) volumes of material elements are related through

A/[A.sub.0] = [[Lambda].sub.2][[Lambda].sub.3] (1)

V/[V.sub.0] = [[Lambda].sub.1][[Lambda].sub.2][[Lambda].sub.3] = A/[A.sub.0] [[Lambda].sub.1] (2)

For the small stretch at yield, [[Lambda].sub.1] [approximately equal to] 1.06, the stretches are related to the normal (small) strains through [[Lambda].sub.i] = 1 + [[Epsilon].sub.i], i = 1, 2, 3, and the volume ratio is related to the (small) volumetric strain [e.sub.v] = [[Epsilon].sub.1] + [[Epsilon].sub.2] + [[Epsilon].sub.3] through v/[v.sub.0] = 1 + [e.sub.v]. Assuming material isotropy and a Poisson's ratio of v,

[e.sub.v] = (1 - 2v)[[Epsilon].sub.1] = (1 - 2v)([[Lambda].sub.1] - 1) (3)

so that

[Sigma] = P/A = P/[A.sub.0] [A.sub.0]/A = P/[A.sub.0] [V.sub.0]/V [[Lambda].sub.1] = [[Lambda].sub.1]/1 + (1 - 2v)([[Lambda].sub.1] - 1) [[Sigma].sub.n] (4)

which, for small strains, reduces to

[Sigma] [approximately equal to] (1 + 2v[[Epsilon].sub.1])[[Sigma].sub.n] (5)

Clearly, [Sigma]= [[Lambda].sub.1][[Sigma].sub.n] for an incompressible material (v = 0.5). The Poisson's ratio for polymers is on the order of 0.4. For a stretch of [[Lambda].sub.1] = 1.06, the values of [Sigma]/[[Sigma].sub.n] for Poisson's ratios of 0.3, 0.35, 0.4, and 0.45 are 1.035, 1.041, 1.047, and 1.054, respectively. Thus, the true stress is on the order of 5% higher than the nominal stress.

CREEP AT HIGH LOADS

In a constant-displacement-rate tensile test, a critical stress [[Sigma].sub.0] is required for initiating neck formation, after which the stress drops to [[Sigma].sub.d]. This drop in stress raises an interesting question: What happens if the stress [Sigma] is maintained at a constant value [[Sigma].sub.c] between [[Sigma].sub.d] and [[Sigma].sub.0], as schematically shown in Fig. 7? (The curve for the constant-displacement-rate test is indicated by a dashed line.)

To answer this question, tests were conducted in which the nominal stress [[Sigma].sub.n] on the specimen was increased linearly with time to a prescribed value [[Sigma].sub.c] ([[Sigma].sub.d] [less than] [[Sigma].sub.c] [less than] [[Sigma].sub.0]), at which it was held constant [ILLUSTRATION FOR FIGURE 7 OMITTED]. The time [t.sub.c] to reach the stress [[Sigma].sub.c] was adjusted to provide a nominally constant loading rate. In the preliminary tests described in this paper, only the longitudinal stretch [[Lambda].sub.1] (longitudinal strain [[Epsilon].sub.1] = [[Lambda].sub.1] 1) was monitored as a function of time. The results of these tests show that the constant tensile stress causes the material to creep. When the stretch approaches [[Lambda].sub.1] = 1.06, the material undergoes very rapid extension resulting in failure. What happens, of course, is that the cross-sectional area of the material decreases with increasing longitudinal stretch caused by creep, so that while the applied load is constant, the true stress continues to increase. It appears that the material necks at [[Lambda].sub.1] [approximately equal to] 1.06, when the true critical stress needed to initiate necking is achieved. On necking, the stress required to propagate the neck decreases. However, because the test is done under load control, the machine increases the cross-head displacement rate in an effort to maintain the prescribed load; the very rapid extension of the specimen results in failure. The higher the value of [[Sigma].sub.c], the higher the creep rate, and the smaller the time at which the material undergoes stable necking.

To confirm this hypothesis, the test procedure was modified to prevent the failure of the specimens caused by the uncontrolled increase in the cross-head displacement rate. Specimens were loaded at a constant displacement rate to a predetermined nominal stress [[Sigma].sub.c], at which it was held constant (just as in the previous tests). To prevent uncontrolled displacement after load drop-off, the machine was programmed to change to displacement control once a prescribed displacement - chosen to ensure homogeneous stretches of about 1.1 - was attained. As in the previous tests, the stretch was monitored by means of a 12.7-mm (0.5-in) gauge-length extensometer. In two tests, standard 2.9-mm-(0.114-in-) thick ASTM D638 specimens were pulled in tension at a nominal stretch rate of [10.sup.-2] [s.sup.-1]; [[Sigma].sub.c] was chosen such that ([[Sigma].sub.0] - [[Sigma].sub.c])/([[Sigma].sub.0] - [[Sigma].sub.d]) had values of 0.25 and 0.5. Both the specimens underwent creep at the constant loads. During the creep deformation, the stretch rate was approximately constant at about 3.5 x [10.sup.-4] [s.sup.-1] and [10.sup.-5] [s.sup.-1] for ([[Sigma].sub.0] - [[Sigma].sub.c])/([[Sigma].sub.0] - [[Sigma].sub.d]) = 0.25 and 0.5 (i.e., for the higher and lower constant loads), respectively, until a stretch of about 1.07, when the creep rate began to increase rapidly with the formation of shear bands.

The strain history for the higher load case is shown in Fig. 8; the strain rate began to increase at about 70 s when the stretch was about 1.075. The displacement cutoff was activated just before 80 s. By then a neck had formed outside the gauge length of the extensometer; the homogeneous deformation inside the gauge length had a stretch of 1.08.

It took about an hour for the displacement cutoff to be activated at the lower load. Because of the low creep rate, it was possible to visually observe the formation of shear bands. Once shear bands appeared, more of them initiated with time as the stretch in the gauge length increased. One shear band began to widen; this appeared to coincide with an increase in the creep rate, resulting in the displacement cutoff being activated when the extensometer showed a homogeneous stretch of 1.09. At this point a wide shear band had formed.

These exploratory tests show that creep deformation eventually results in strain localization (stable neck formation) at stretches in the range of 1.06 to 1.09. However, the deformation and stress state at which shear bands first appear have not been characterized. The simultaneous monitoring of the stretch in the longitudinal (as in the above experiments) and thickness or width directions would make it possible to monitor both the homogeneous stretch and the true stress in the specimen, thereby helping to better characterize the onset of yield.

Tests were also done at stresses lower than [[Sigma].sub.d], for which the creep rate was very low. In many cases a crack initiates, which eventually results in brittle fracture caused by crack growth. Also, depending on the magnitude of the load, the material can craze.

UNIAXIAL EXTENSION OF WIDE SPECIMENS

The tensile tests described so far were conducted on thin specimens with a standard width of 12.7 mm (0.5 in) over the gauge length. In these narrow specimens the edges are not constrained, so that the stress is essentially one-dimensional, and the lateral strains are related to the longitudinal strain through [[Epsilon].sub.2] = [[Epsilon].sub.3] = - v[[Epsilon].sub.1]. In very wide specimens, the state of stress close to the edges should essentially be one-dimensional. However, away from the edges, the material is constrained from moving in the width direction, resulting in a biaxial stress state in that region.

In the experiments to be described, a special set of grips was used for stretching PC specimens with widths up to 100 mm (4 in). After marking a square grid, each specimen was stretched at a constant displacement rate. The deformation of the specimen was recorded by using a video camera; a macro lens was used to obtain closeup views. First, a homogeneous stretching of the material was observed. Then, at a critical load, a shear band appeared across the specimen, as schematically shown in Fig. 9a. On further extension the shear band became wider [ILLUSTRATION FOR FIGURE 9B OMITTED]; this widening of the shear band does not occur in metals.

At some stage another shear band initiated, as shown by the dashed line in Fig. 9b. On further stretching, the second shear band became progressively wider, leaving behind islands of unyielded material [ILLUSTRATION FOR FIGURE 9C OMITTED]. While the first shear band rotated longitudinal grid lines (schematically shown by straight lines in Figs. 9a to d) to the right [ILLUSTRATION FOR FIGURE 9B OMITTED], the second shear band caused these lines to rotate back [ILLUSTRATION FOR FIGURE 9C OMITTED], until all the lines were again aligned longitudinally [ILLUSTRATION FOR FIGURE 9D OMITTED]. Throughout this yield history, no "plane stress" effects - reduction in the distance between the longitudinal lines - were observed, neither during the initial shearing [ILLUSTRATION FOR FIGURE 9B OMITTED] nor during the reverse shearing [ILLUSTRATION FOR FIGURE 9C OMITTED] Thus, the transition from the homogeneous, deformed state to the yielded state occurs via shear in which there are no edge effects. On unloading, the longitudinal stretch in the yielded material was on the order of [[Lambda].sub.1] = 1.7, while the stretches in the width and thickness directions were measured to be about [[Lambda].sub.2] [approximately equal to] [[Lambda].sub.3] [approximately equal to] 0.7.

These experiments were conducted on specimens of different thicknesses (1.5, 3, and 6.35 mm; 0.06, 0.12, and 0.25 in) and widths (12.7, 25.4, 50.8, and 101.6 mm; 0.5, 1, 2, and 4 in). In each case, the transition from homogeneously deformed material at a stretch of [[Lambda].sub.1] = 1.06 to the oriented material with a stretch [[Lambda].sub.1] [approximately equal to] 1.7 occurred through shear bands that coalesced to form a neck that on further extension propagated along the specimen in a stable manner. However, the shear bands formed necks in different ways. For example, in some cases, two symmetric shear bands formed at the same instant, and neck formation occurred in a symmetric manner.

These experiments on the tensile extension of thin rectangular specimens clearly show that the material can be homogeneously stretched only up to a limiting stretch of [[Lambda].sub.1] [approximately equal to] 1.06. Further extension results in strain localization in the form of shear bands in which the material has undergone very large shear. Although the material is still being pulled in the longitudinal direction, the state of stress in the shear bands is not uniaxial. The two sets of shear bands, each of which induces finite shear in the material, coalesce to move the sheared material back to a state in which it has undergone a stretch [[Lambda].sub.1] [approximately equal to] 1.7 in the longitudinal direction and stretches [[Lambda].sub.2] [approximately equal to] [[Lambda].sub.3] = 0.7 in the lateral directions. This doubly sheared region constitutes the neck that is formed and consists of highly oriented material. On further extension of the specimen, the necked material does not stretch any further. Rather, more of the homogeneously deformed unnecked material orients, and the neck propagates along the specimen in a stable manner under a constant uniaxial load. Finally, when the neck reaches the wider shoulder of the specimen, the load required to stretch the specimen at a constant displacement rate increases, and the necked material is stretched beyond [[Lambda].sub.1] = 1.7.

The deformation history of the material where the shear bands initiate is different from the deformation histories of the rest of the material. The shear bands are formed at an initial nominal stress [[Sigma].sub.0], after which the stress drops to a draw stress [[Sigma].sub.d] at which the neck propagates along the specimen. Thus, in the rest of the material, the stretch first builds up to [[Lambda].sub.1] [approximately equal to] 1.06. Then, a reduction of the stress from [[Sigma].sub.0] to [[Sigma].sub.d] [approximately equal to] 0.75 [[Sigma].sub.0] causes the stretch to decrease to [[Lambda].sub.1] [approximately equal to] 1.025 - the actual value depends on the strain rate and the temperature. The material is maintained at this reduced stretch until the advancing neck passes through, increasing the stretch to [[Lambda].sub.1] = 1.7.

Theocaris and Hadjiiossiph (33) used the moire technique to accurately map the evolution of a neck in a PC specimen having a 3.2 x 30-mm rectangular cross section. (The term neck was used for regions in which strain localization had occurred.) A shear band was found to initiate at an angle of 55 [degrees] with the longitudinal axis of the specimen. Although the development of a fully developed neck was ascribed to a coalescence of opposing shear bands, the strain field was only mapped for a single, widening shear band, corresponding to the deformation mode schematically shown in Fig. 9b. Thus, the deformation regime in which sheared material [ILLUSTRATION FOR FIGURE 9B OMITTED] is rotated back along the longitudinal direction [ILLUSTRATION FOR FIGURE 9C OMITTED], was not studied.

BIAXIAL STRETCHING OF CLAMPED CIRCULAR SHEETS BY LATERAL FLUID PRESSURE

To map the phenomenology of biaxial deformation in polymers, a special bulge tester was fabricated in which clamped circular sheets of polymer can be stretched by a controlled laterally applied fluid pressure. The apparatus schematically shown in Fig. 10 is similar to the hydraulic bulge tester described by Young et at (34). A similar device was used by Kirkland et al. (35) to obtain biaxial stress-strain curves for cellulose nitrate and acrylonitrile-butadiene-styrene. A circular sheet specimen is clamped by means of a metal ring onto a steel plate with a circular hole. Fluid pressure can be applied by displacing a servohydraulically controlled piston. Tests can be performed under controlled pressure or by controlling the amount of fluid forced against circular sheet specimens.

In the preliminary tests described herein, 1.5-mm-(0.06-in-) thick circular sheet specimens of PC were clamped to provide circular test regions with radii of R = 101.5 mm (4 in). To facilitate measurement of the deformation, a grid formed by concentric circles at radial intervals of 6.35 mm (0.25 in) and eight radial straight lines at intervals of 45 [degrees] were marked on each circular specimen. These specimens were loaded by oil that was pressurized by the controlled motion of a servohydraulically controlled piston.

During each experiment, the applied pressure p and the displacement of the disk at its center (dome height) were monitored as functions of time, the latter by means of an LVDT. In this series of tests, sheet specimens were deformed to predetermined dome heights at which the specimens were unloaded by releasing the oil pressure. Five specimens were loaded to different final pressures at the same volumetric rate, for which the pressure-time and the pressure-dome-displacement histories are shown in Figs. 11 and 12, respectively. The curves for the four lower pressures form a part of the curve for the highest pressure. Three regimes can be identified: First, in the initial stage the pressure-displacement curve has a small slope; then the slope increases to an approximately constant value; finally, the slope decreases sharply to a lower (approximately) constant slope.

In these tests, the servohydraulically controlled piston was moved at a constant displacement rate to force oil against the clamped sheet at a constant volumetric rate of 7.72 [cm.sup.3]/s. The piston was programmed to stop when a prescribed pressure was attained, and the piston was maintained at this position for some time. Because of creep at these high pressures, the pressure in the constant volume of oil actually drops continually, first rapidly, and then more slowly, as can be seen from the pressure-time traces in Fig. 11. The increase in the dome height caused by creep is rather small because of the low compressibility of the oil - a small increase in the volume under the dome causes a rapid decrease in the pressure, resulting in reduced creep. This small increase in the dome height appears as a slight right-ward bulge in the unloading curves in Fig. 12. The pressure was released after some time resulting in some elastic recovery, as shown by the decrease in the dome height along the unloading curves in Fig. 12.

The measured shapes of the domes after unloading are shown in Fig. 13. The change in the curvature of the deflected shape near the clamped edge shows that bending effects are important at the clamped circular boundary. Note that the clamping rings did not have sharp edges; the edges were rounded to ensure that excessive local stresses would not cause failure.

As the pressure under the sheet was increased, the sheet deformed continually in the shape of a dome. However, at some stage in the deformation, strain localization occurred near the clamped edge but away from it. On further deformation, the localized region took on a V-shape aligned with the radial direction, with the vertex of the V pointing radially outward, as shown in Fig. 14. This type of strain localization was first reported by Lege (36), who showed that a similar bulge test on sheets of poly(ethylene terephthalate) resulted in strain localization over much larger regions of the resulting dome.

As mentioned earlier, a grid, consisting of concentric circles with the radii increasing in steps of 6.35 mm (0.25 in) and of eight radial lines at intervals of 45 [degrees], was marked on each circular specimen prior to each test. The variation of the stretch in the unloaded dome was determined as follows: The arc length between consecutive deformed circles was determined along each radial line by marking the location of the deformed circles on a strip of paper placed along a radial line, and by then measuring the distance between these marked intervals. The radial stretch [[Lambda].sub.1] was then calculated by dividing the deformed length by the original length of each radial interval. The hoop stretch [[Lambda].sub.2] = [r.sub.d]/r was calculated from a measurement of the deformed radius [r.sub.d] of an initial circle of radius r. Also the stretch in the thickness direction [[Lambda].sub.3] = [t.sub.d]/[t.sub.0] was obtained by measuring the local deformed thickness, [t.sub.d], at each grid location by means of a Hall-effect device (Magna-Mike Model 8000), where to is the initial sheet thickness. Because of the techniques used, the accuracy of stretch measurement was highest for [[Lambda].sub.3] and lowest for [[Lambda].sub.1].

The variations of the radial, hoop, and thickness stretches along the radius are shown, respectively, in Figs. 15, 16, and 17 for the specimens for which the pressure-time and pressure-displacement histories are shown, respectively, in Figs. 11 and 12. Note that for larger radii, the curves for the thickness stretch [[Lambda].sub.3] are based on thickness measurements on the thicker regions between the consecutive Vs. The thickness stretch inside each V is much smaller. Clearly, the curves for [[Lambda].sub.3] are the smoothest, while those for [[Lambda].sub.1] are the least smooth. The dip in the thickness stretch near r/R = 1 is caused by bending effects near the clamped edge. Because of symmetry, [[Lambda].sub.1] should equal [[Lambda].sub.2] at r/R = 0, and the data in Figs. 15 and 16 confirm this. If the dome were spherical, then [[Lambda].sub.1] = [[Lambda].sub.2] and [[Lambda].sub.3] would not vary over the specimen. The approximately zero slopes of the [[Lambda].sub.1], [[Lambda].sub.2], and [[Lambda].sub.3] curves in the neighborhood of r/R = 0 show that the domes have spherical shapes only near r/R = 0. The shapes of the [[Lambda].sub.3] curves show that with increasing deformation (increasing dome heights), the equibiaxially stretched (spherical) region increases. This increase is important because a measurement of the curvature of this region, together with a simultaneous measurement of the pressure, can be used to obtain stress-stretch relations under biaxial deformations. For larger values of r/R, the thickness stretch decreases (thickness reduction decreases) continuously. The measured variations of the radial and hoop stretches are compared in Fig. 18. These data indicate that [[Lambda].sub.1] [less than] [[Lambda].sub.2] near the clamped edge where the stretches are low. For higher stretches (smaller radii) [[Lambda].sub.1] approaches [[Lambda].sub.2].

Although the deformation increased monotonically with increases in pressure in most of the clamped sheet - right through yield into the post-yield regime - strain localization occurred near the clamped edges resulting in V-shaped depressions; these depressions were aligned radially, with the apex of the V pointing outward [ILLUSTRATION FOR FIGURE 14B, C, D OMITTED]. The deformations in the bulk of the deformed sheet merged continuously across the mouth of the V into its interior. However, the thickness jumped sharply across the sides of the V. For example, for the largest dome (solid squares in Fig. 17), the thickness stretch at r/R = 0.95 dropped from a value of [[Lambda].sub.3] [approximately equal to] 1.0 in the homogeneously deformed region to [[Lambda].sub.3] [approximately equal to] 0.75 within the V. Clearly, the strain localization in these highly deformed regions is analogous to the strain localization that results in the formation of the stably propagating neck in a tensile test.

The results of these biaxial stretch tests show a clear qualitative difference from the phenomenology of yield in a uniaxial tensile test: The transition from the unyielded to the yielded material in a tensile test occurs through strain localization in the form of shear bands, via a mechanism in which the strain field is no longer one-dimensional, through an almost discontinuous jump in the stretch from [[Lambda].sub.1] = 1.05 to [[Lambda].sub.1] [approximately equal to] 1.7. In contrast, under biaxial tension, the material deforms continuously from an unyielded to a yielded state; both [[Lambda].sub.1] and [[Lambda].sub.2] increase monotonically, while [[Lambda].sub.3] decreases monotonically. Interestingly, strain localization only occurs close to r/R = 1.

THERMALLY INDUCED RECOVERY FROM MECHANICALLY YIELDED STATE

Polymers are known to "remember" the original shape from which they have been deformed. On subsequent heating to the glass transition temperature ([T.sub.g] [approximately equal to] 150 [degrees] C, 302 [degrees] F, for PC), they are known to recover to the original undeformed state. To study this effect, PC specimens were first stretched in tensile tests to a yielded state; the necked portions were then heated to different temperatures below [T.sub.g] during which the recovery of the necked material was recorded.

Several 1.5-mm-(0.06-in-) thick ASTM D638 bars with 152.5-mm-(6-in-) nominal lengths were stretched in tensile tests to (unloaded) lengths of 197 mm (7.75 in). These specimens had approximately 108-mm-(4.25-in-) long yielded, stably necked regions. The necked material, cut from these specimens, was clamped in the grips such that the material between the grips corresponded to an original (before the specimen was stretched) length of 50.8 mm (2 in); the clamped specimen and the grips were enclosed inside a temperature-controlled oven. The purpose of the experiments was to map the recovery of the specimens as a function of its temperature under no-load conditions. Ideally, the specimen should be heated in a load-controlled test with the load set to zero. Because of limits on load resolution, the specimen will always be subjected to a small load. Also, an increase in temperature will cause the specimen to expand. To prevent any buckling, the no-load setting for the load-control test was carefully set at [0.sup.+]. The temperature of the specimen was monitored by a thermocouple attached to its surface. The oven temperature was then raised and held at different temperatures (120, 125, 130, 138, 146, and 150 [degrees] C) below the glass transition temperature.

For these transient tests, Fig. 19 shows the variations with time of the nondimensional specimen temperature, (T - [T.sub.a])/([T.sub.g] - [T.sub.a]), where [T.sub.a] is the ambient temperature, and the nondimensional recovery ([l.sub.i] - l)/([l.sub.i] - [l.sub.f]) of the length, where [l.sub.i] and [l.sub.f] are the initial and final lengths of the necked material. These tests show that the material recovers most of the "permanent" deformation at temperatures close to [T.sub.g]. However, substantial recovery begins at temperatures well below [T.sub.g], as can be seen from the recovery curve for 120 [degrees] C. (In these tests, heat transfer at the grips results in lower temperatures in the material close to the grips, so that the curves in this figure underestimate the final recovery.) While these tests show that the material begins to recover from "permanent" deformation below [T.sub.g], a one-to-one comparison of the recovery with the temperature would not be meaningful because of the transient temperature rise; such a correlation would not account for isothermal, time-dependent recovery.

These preliminary results clearly show that the necked (oriented) material begins to recover from its deformed state at temperatures well below [T.sub.g]. However, the total recovery in one hour at temperatures lower than about 140 [degrees] C is less than about 20%. Also, the rate of recovery at these low temperatures is very small after a steady temperature has been attained. Close to the glass transition temperature, the material appears to rapidly recover the entire deformation. The results of these experiments raise interesting questions. Is there just one thermally activated recovery mechanism that accelerates the recovery process as the temperature approaches [T.sub.g], or are the recovery mechanisms different for high and low temperatures? At the lower temperatures (see the curves for 120 to 138 [degrees] C) does the amount of recovery level off (zero recovery rate) or does the recovery continue at a slow rate?

CONCLUDING REMARKS

The phenomenology of yield in bisphenol-A polycarbonate has been explored through four types of tests: a series of tensile tests on thin, rectangular standard ASTM D638 dog-bone specimens; tensile tests on thin, wide rectangular specimens; biaxial "bulge" tests in which thin, clamped circular disks were stretched by lateral pressure; and temperature-induced recovery tests, in which a yielded specimen was heated to the glass transition temperature. A clearer picture of the deformation behavior of this material has emerged from these tests, which confirm that the complex deformation phenomenology of this material is qualitatively very different from that of metals.

As is well known, when a thin, narrow rectangular cross-sectioned dog-bone specimen is stretched at a constant displacement rate in a tensile test, the nominal stress [[Sigma].sub.n] increases to a critical value [[Sigma].sub.0] at a stretch of [[Sigma].sub.0] [approximately equal to] 1.06. Then the stress falls off precipitously to a lower value [[Sigma].sub.d], the draw stress, at which the stretch in the material is on the order of [[Lambda].sub.d] [approximately equal to] 1.7. The transition from the homogeneously deformed material at [[Lambda].sub.0] [approximately equal to] 1.06 to the "necked" material with a homogeneous stretch of [[Lambda].sub.d] [approximately equal to] 1.7 occurs extremely fast, almost instantaneously when the material is being stretched at a nominal strain rate of [10.sup.-2] [s.sup.-1]. Because the necked material is stiffer and "stronger" than the unnecked homogeneously deformed material, further extensions of the specimen result in a stable neck propagation along the specimen, which causes the stretch in the unnecked material to change from [[Lambda].sub.0] [approximately equal to] 1.06 to [[Lambda].sub.d] [approximately equal to] 1.7 over very short transition zones. During this neck propagation, the stretch in the necked, or drawn, material remains constant at [[Lambda].sub.d] [approximately equal to] 1.7 until the neck reaches the wider portion of the specimen, after which the nominal stress in the material increases, resulting in an increase in stretch beyond [Lambda] = 1.7, finally causing failure in the necked material.

The experiments described in this paper have shown that while both [[Lambda].sub.0] and [[Lambda].sub.d] increase with increasing strain rates in the range of [10.sup.-4] to [10.sup.0] [s.sup.-1] and with decreasing temperatures in the range of 65 to 22 [degrees] C, their ratio is approximately constant at [[Sigma].sub.d]/[[Sigma].sub.0] = 0.75. Tests where the specimens were first stretched at a linearly increasing stress rate to a stress [Sigma], [[Sigma].sub.d] [less than] [Sigma] [less than] [[Sigma].sub.0], and then were maintained at this stress showed that the material creeps homogeneously until a stretch of [Lambda] [approximately equal to] 1.06 is attained, at which the material undergoes strain localization resulting in the formation of a neck. The time required to attain this critical stretch decreases with increasing [Sigma]. Preliminary tests would seem to indicate that for [Sigma] [less than] [[Sigma].sub.d], the material creeps progressively more slowly with decreasing [Sigma] and finally fails in a brittle manner by crack growth around flaws.

Experiments on tensile stretching of thin, wide rectangular specimens have shown that the transition from the unnecked homogeneously deformed material at [Lambda] = 1.06 to the homogeneously deformed material at [Lambda] [approximately equal to] 1.07 occurs through a complex strain localization phenomenon. At [Lambda] [approximately equal to] 1.06, a shear band suddenly nucleates at one edge and propagates across the width of the specimen; this band widens with further extension of the specimen. The material in this region has undergone large shear that remains constant as the band widens. At some stage, a reverse shear band nucleates, widens, and propagates across the width. The intersection of the two shear bands causes the sheared material to rotate back and to align along the direction of extension. Alternatively, two intersecting shear bands can nucleate simultaneously, resulting in a more symmetric evolution of the yielded neck. Clearly, then, the transition from [Lambda] [approximately equal to] 1.06 to [Lambda] [approximately equal to] 1.7 in a tensile test does not occur through a one-dimensional deformation or stress field but, rather, through at least a two-dimensional, if not a fully three-dimensional field. Therefore, a one-dimensional stress-stretch curve should really be represented as a full curve from 0 [less than or equal to] [Lambda] [less than or equal to] [[Lambda].sub.0] [approximately equal to] 1.06 and for [Lambda] [greater than or equal to] [[Lambda].sub.d] [approximately equal to] 1.7. The region [[Lambda].sub.0] [less than or equal to] [Lambda] [less than or equal to] [[Lambda].sub.d] cannot be achieved in a one-dimensional tensile test.

Biaxial stretching experiments, in which a circular sheet clamped at the edges was expanded in a "bulge" mode by forcing oil against the sheet at a constant volumetric rate, showed a different yielding mode: In contrast to the jump in the stretch from [Lambda] [approximately equal to] 1.06 to 1.7 in a tensile test, the stretches in the radial, hoop, and thickness directions [[Lambda].sub.1], [[Lambda].sub.2], and [[Lambda].sub.3], respectively, varied continuously as the pressure under the sheet increased; [[Lambda].sub.1] and [[Lambda].sub.2] increased continuously from 1.0 to about 1.65, while [[Lambda].sub.3] decreased continuously from 1.0 to about 0.7. As the deformations near the outer edges of the disk increased, strain localization did occur near, but away from, the clamped edge. With further deformation, the "shear band" expanded in the form of a V-shaped region, with the apex pointing radially outward. The interior of the V had undergone very large stretches that continuously merged, in the radial direction, with the large deformations in the homogeneously yielded regions across the open end of the V. Very large changes in (thickness) stretch were measured across the sides of the V. A comparison of these results with those from tensile tests shows that whether or not yield occurs through strain localization depends on the state of deformation, or stress. Based on these limited experiments, it is tempting to propose that strain localization is likely to occur only in near one-dimensional deformation fields.

Amorphous polymers are known to be strongly history-dependent materials; they "remember" the deformations to which they have been subjected. Temperature-induced recovery of deformation caused by yield was studied by heating stably necked regions of bars stretched in tension. The tests showed that while most of the recovery occurs near the glass transition temperature [T.sub.g] of the material, the specimens begin to recover from the yielded state well below [T.sub.g].

ACKNOWLEDGMENTS

The authors thank Donald F. Mowbray for having supported this project over several years, and for having provided encouragement when it was most necessary. The inputs of several people, who contributed to this effort, are gratefully acknowledged: Roger N. Johnson supervised the design and fabrication of the biaxial sheet stretching apparatus with inputs from Horst deLorenzi and Louis P. Inzinna. L. P. Inzinna made major contributions that were crucial to the success of the program; he "debugged" and instrumented the apparatus, developed the test procedures used, and did all the biaxial sheet stretching tests. L. P. Inzinna and Linda A. Briel did all the tensile tests, and reduced all the data in this paper.

Special thanks are due to Julia A. Kinloch for her help and patience during the preparation of this paper.

REFERENCES

1. I. Marshall and A. B. Thompson, Proc. R. Soc., A, 221, 541 (1954).

2. P. I. Vincent, Polymer, 1, 7 (1960).

3. R. E. Robertson, J. Appl. Polym. Sci., 7, 443 (1963).

4. V. K. Stokes and H. F. Nied, J. Eng. Mater. Technol., 108, 107 (1986).

5. H. F. Nied and V. K. Stokes, J. Eng. Mater. Technol., 108, 113 (1986).

6. R. P. Kambour, J. Polym. Sci., D, 7, 1 (1973).

7. R. P. Kambour, "Crazing," in Encyclopedia of Polymer Science and Engineering, 2nd ed., pp. 299-322, Mark, Bikales, Overberger, and Menges, eds., John Wiley and Sons, New York (1986).

8. W. Whitney and R. D. Andrews, J. Polym. Sci., C, 16, 2981 (1967).

9. C. A. Pampillo and L. A. Davis, J. Appl. Phys., 42, 4674 (1971).

10. I. M. Ward, J. Mater. Sci., 6, 1397 (1971).

11. K. D. Pae, J. Mater. Sci., 12, 1209 (1977).

12. P. B. Bowden, "The Yield Behavior of Glassy Polymers," in The Physics of Glassy Polymers, pp. 279-339, R. N. Haward, ed., John Wiley and Sons, New York (1973).

13. R. N. Haward, "The Post-Yield Behavior of Amorphous Plastics," in The Physics of Glassy Polymers, pp. 340-93, R. N. Haward, ed., John Wiley and Sons, New York (1973).

14. R. N. Haward, ed., The Physics of Glassy Polymers, John Wiley and Sons, New York (1973).

15. C. G'Sell and J. J. Jonas, J. Mater. Sci., 14, 583 (1979).

16. M. C. Boyce and E. M. Arruda, Polym. Eng. Sci., 30, 1288 (1990).

17. J. Amoedo and D. Lee, Polym. Eng. Sci., 32, 1055 (1992).

18. C. G'Sell and J. J. Jonas, J. Mater. Sci., 16, 1956 (1981).

19. C. G'Sell, N. A. Aly-Helal, and J. J. Jonas, J. Mater. Sci., 18, 1731 (1983).

20. C. G'Sell, J. M. Hiver, A. Dahoun, and A. Souahi, J. Mater. Sci., 27, 5031 (1992).

21. G. Buisson and K. Ravi-Chandar, Polymer, 31, 2071 (1990).

22. G. Titomanlio and G. Rizzo, Polymer, 21, 461 (1980).

23. E. M. Arruda, M. C. Boyce, and H. Quintus-Bosz, "Anisotropic Effects on the Finite Strain Deformation Behavior of Glassy Polymers," in Plastics and Plastic Composites: Material Properties, Part Performance, and Process Simulation, MD-Vol. 29, pp. 1-12, V. K. Stokes, ed., American Society of Mechanical Engineers, New York (1991).

24. W. Wu and A. P. L. Turner, J. Polym. Sci., B, 11, 2199 (1973).

25. C. G'Sell, S. Boni, and S. Shrivastava, J. Mater. Sci., 18, 903 (1983).

26. W. A. Spitzig and O. Richmond, Polym. Eng. Sci., 19, 1129 (1979).

27. R. M. Caddell and J. W. Kim, Int. J. Mech. Sci., 23, 99 (1981).

28. A. F. Yee and L. M. Carapellucci, SPE ANTEC Tech. Papers, 29, 370 (1983).

29. S. S. Sternstein, L. Ongchin, and A. Silverman, Appl. Polym. Symp., 7, 175 (1968).

30. S. S. Sternstein and L. Ongchin, Polymer Preprints, American Chemical Society, 10 (2), 1117 (1968).

31. S. S. Sternstein and F. A. Myers, J. Macromol. Sci.-Phys., B8 (3-4), 539 (1973).

32. A. F. Yee and P. D. Detorres, Polym. Eng. Sci., 14, 691 (1974).

33. P. S. Theocaris and C. Hadjiiossiph, Eng. Fract. Mech., 12, 241 (1979).

34. R. F. Young, J. E. Bird, and J. L. Duncan, J. Appl. Metal Working, 2, 11 (1981).

35. J. T. Kirkland, J. L. Duncan, and R. N. Haward, Polymer, 11, 562 (1970).

36. D. J. Lege, Alcoa Laboratories Centennial Technical Seminar, "Fundamentals of Deformation Processes," Hilton Head Island, S.C., personal communication (1987).

37. J. M. Powers and R. M. Caddell, Polym. Eng. Sci., 12, 432 (1972).