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Uniaxial, shear, and Poisson relaxation and their conversion to bulk relaxation: studies on poly(methyl methacrylate).


Within the context of linearly viscoelastic behavior, it is sufficient for any analysis effort to define any two of the four material functions (in relaxation or creep) describing uniaxial, shear, bulk, or Poisson response. In the relaxation mode these properties are identified as the uniaxial modulus E(t), the shear modulus [Mu](t), the bulk modulus K(t), and Poisson's ratio v(t), with appropriate functions defined for the creep complement. By implication any two can then be computed from the others.

In current engineering practice it is common to deal with either the uniaxial or shear relaxation modulus, assuming that the other function is described sufficiently well in terms of a constant Poisson's ratio or a constant bulk modulus. With the advent of more refined and powerful methods of computation the need arises to know the material behavior with an increasing degree of sophistication. As a consequence, it becomes important to be able to determine the material properties with more precision for a full characterization of (even) linearly viscoelastic materials than current engineering practices allow. For example, few reliable records of two of the four time-dependent material functions for the same materials exist. Similarly, there is a dearth of information on the time dependent bulk response of linearly viscoelastic materials (in the transition range). These difficulties in understanding viscoelastic materials more fully derive from the often poorly appreciated time dependent characteristics of these materials, their temperature sensitivities, and the fact that both large and small deformations may be involved and need to be addressed separately.

While some experimentally oriented investigators have realized serious limitations in being able to determine any of the material functions from the others (1), this recognition is anything but general, and it appears appropriate to demonstrate the experimental limitation underlying this conversion failure. Moreover, even though a purely analytical demonstration using typical, though fictitious, properties might suffice, it appears more convincing to many to deal with measured material properties. As a result we present here data accumulated gradually in our laboratory over several years that are likely to be informative in their own right, while, in addition, demonstrating the problem of interconversion failure.

We report two groups of material functions for poly(methyl methacrylate) (PMMA). For the first group of tests, strip specimens are used to measure the uniaxial modulus and Poisson's ratio in relaxation with an image moire method on a Rohm & Haas material. In the second group, tubular specimens are used to measure the uniaxial and the shear relaxation moduli of an ACE material as monitored with the digital image correlation method (2). Moreover, preliminary uniaxial tension and Poisson's ratio measurements conducted dynamically at a frequency of 5 MHz, and uniaxial tension experiments at a strain rate of 0.1%/min at 22 D C; i.e., 83 [degrees] C below the glass transition, suggested that the uniaxial modulus still decreases rather significantly under both conditions, so that relaxation behavior at lower than room temperature conditions was indicated. Accordingly, temperatures as low as -40 [degrees] C are included in our study.

The earliest comprehensive temperature-dependent data for E(t) on PMMA were reported by McLoughlin and Tobolsky (3), but similar shear relaxation data do not appear to have been collected over as wide a temperature range. Several investigators have contributed to the measurement of E(t) for PMMA, among them, Rusch (4) provided relaxation data between -20 [degrees] C and 106 [degrees] C. The temperature and time dependent Poisson's ratio, v(t), has not been measured directly in the transition range, though Poisson's ratio in the "glassy" state has been measured repeatedly as a (limited) function of time: Gilmour et al. (5) summarized results on E and v for polystyrene; Wilson et al. (6, 7) measured v(t) on polyethylene sheets using both an optical technique and Michelson interferometry; McCammond et al. (8) determined v(t) from the deflection of a freely supported circular plate under uniformly distributed pressure for PMMA and PVC materials at room temperature. However, the experimental methods employed in these studies, except possibly the optical ones (6, 7), lead to serious difficulties when applied over a wide range of temperatures.


The analytical relations connecting the various material functions of linear viscoelasticity are readily derived and need not be recounted here. We document here only the formulae for the numerical computations that are based on the Hopkins and Hamming algorithm (9). The range of material response is divided into "n" (non-constant) time intervals [t.sub.n] so that the relaxation bulk modulus can be computed as a function of time from the uniaxial modulus E(t) and the Poisson function v(t) via

[Mathematical Expression Omitted] (1)


K([t.sub.1]) = E([t.sub.1])/3 + [K.sub.g][[v.sub.g] - v(t.sub.1])] / 1 - 3[v.sub.g] + [t.sub.1]

and [K.sub.g] = [E.sub.g]/3(1 - 2[v.sub.g]), [E.sub.g] = E(0), [v.sub.g] = v(0); on the other hand, given E(t) and the shear modulus [Mu](t) this quantity is computed from

[Mathematical Expression Omitted] (2)


f(t) [equivalent to] [integral of] [3[Mu]([Xi]) - E([Xi])]d[Xi] between limits t and 0, g(t) [equivalent to] [integral of] E([Xi])d[Xi] between limits t and 0,


K([t.sub.1/2]) = [Mu]([t.sub.1/2])g[t.sub.1] / 3f([t.sub.1]).


As mentioned, material functions for PMMA were determined for two variants: the first set led to uniaxial relaxation modulus and Poisson's ratio using a flat specimen for a Rohm & Haas material. The second set of measurements yielded uniaxial and shear relaxation functions as determined through tension and torsion of a hollow cylinder specimen made of an ACE PMMA.

Measurements of the Uniaxial Modulus and Poisson's Ratio

Image moire was used to monitor the specimen deformation with time under strain, and force recording was accomplished on an MTS servo-hydraulic testing system via a Masscomp data acquisition system; the moire fringe data was acquired photographically.

Material and Specimen Preparation

The material for this portion of study (commercial PMMA, Rohm & Haas, nominally 4.0 mm thick and possessing a [T.sub.g] of 105 [degrees] C) was annealed(4) for 2 h in air at 120 [degrees] C and cooled (slowly) at 5 [degrees] C/h to room temperature. The specimen is shown in Fig. 1, with moire grid locations and orientations indicated. Its ends were reinforced with aluminum tabs to reduce grip creep in that portion of the specimen. As a result, very stable [[Epsilon].sub.yy] strains were obtained as determined by continuous monitoring. For each test at the various temperatures new specimens were used to eliminate uncertainties arising from previous strain and thermal histories. All specimens were cut from a single large sheet of PMMA without special orientation preference.

The photoresist method was used to provide moire gratings for monitoring the axial and transverse strains. A master grating with 40 lines per millimeter was found optimal, since a finer grating developed resolution problems due to multiple reflections of the light beam as it passed through the window of the environmental chamber.

Physical Test Arrangement

A schematic of the test setup is given in Fig. 2. The lens system is adjusted to provide an initial, small mismatch between the grating on the specimen and its screen image to enhance the sensitivity of the measurements. This lens system is used to measure the ratio of the transverse strain and axial strain via the moire method.

An MTS servo-hydraulic system was used to load the specimen to a constant strain and the corresponding axial load was recorded by the Masscomp data acquisition system. In addition to the moire monitoring, the strain was tracked by an MTS extensometer with a gauge length of 25.4 mm. A Russell environmental unit provided temperature control between -40 [degrees] C and 125 [degrees] C. The temperature was continuously monitored by a thermocouple close to the specimen, which was always within [+ or -]0.2 [degrees] C of the set temperature. Optically flat window glass in the environmental control housing reduced measurement errors due to light aberrations. Straining by ramp deformation entailed a typical rise time of 1 second. The constant strain was 0.2% at all temperatures except 0.5% at -40 [degrees] C. These small strains were necessary to avoid nonlinear effects. Using rigid wedge grips, the small strain could always be maintained with an accuracy of 1% for the constant strain, [[Epsilon].sub.y 0]. The room humidity remained at [approximately]50% for these tests. The load cell has a full capacity of 15kN as calibrated to a precision of [+ or -]0.5N. Dimension and load levels were such as to allow the relaxation modulus to be determined with an accuracy of [+ or -]4%.

Image Processing and Data Analysis

Fringe images taken at various times (motor driven Nikon F camera with a Nikon 85 mm f 4.5 lens) were scanned with a digitizing scanner possessing a resolution of 118 pixels per cm (the moire pattern covered an area of 4.3 x 10 cm on the photograph). At least 500 line passes were taken across the fringes and the gray scale distribution was averaged to obtain the mean fringe spacing f(t). Accuracy calibration of this method was accomplished with the help of an interferometrically constructed fringe pattern of 3.96 mm. Processing of this image produced a fringe spacing of 3.98 mm. This 0.02 mm difference corresponds to a 0.5% error relative to the known spacing. We accept this error bound for all fringe evaluations.

To evaluate the correspondence for the accuracy in measuring Poisson's ratio one needs to evaluate the error incurred in the moire fringe evaluation. To achieve improved sensitivity of the moire process one employs an initial mismatch between image and reference grating such that an initial fringe density [f.sub.0] results. If p is the pitch of the master (observation) grating and F is the image magnification then

[[Epsilon].sub.xx](t) = [1/[f.sub.x](t) - 1/[f.sub.x 0]] p/F (3)

with Poisson's ratio given by

v(t) = - [[Epsilon].sub.xx](t) / [[Epsilon].sub.y 0] (4)


[[Epsilon].sub.y 0] = [1/[f.sub.y] - 1/[f.sub.y 0]] p/f (5)

so that with [f.sub.x 0] = [f.sub.y 0] = [f.sub.0] and a constant [f.sub.y]

[Mathematical Expression Omitted] (6)

One derives from this a relative error due to the errors in [f.sub.x](t) and [f.sub.y] ([f.sub.o] remains constant for all measurements) as

[Mathematical Expression Omitted] (7)

which interprets into a [+ or -]10% error bound if the denominators are averaged over the range of the experimental data.

Measurements of the Uniaxial and Shear Response on Tubular Specimens

Material and Specimen Preparation

The material for this portion of study was commercially cast PMMA stock of 38 mm diameter (ACE, now a part of Ono, [T.sub.g] = 105 [degrees] C). Cast instead of extruded rods were chosen so as to avoid possible anisotropy due to molecular alignment incurred during processing. Each rod, delivered in 152 cm lengths, was cut into blanks roughly 1 cm longer than the finished specimen length, and the short rods were annealed in a Texaco ISO 46 hydraulic oil bath (boiling point = 355 [degrees] C) at 115 [degrees] C for 4 h and then cooled to room temperature slowly by cutting the power to the temperature chamber; the cooling rate was [approximately]5 [degrees] C/h. This annealing process was necessary to remove the memory of thermal and loading history stored in the material during the casting process.(5) Tubular specimens with an outer diameter of 25.15 mm and an inner diameter of 19.05 mm were machined from these rods. During the machining process, coolant was constantly circulated in order to avoid damage due to overheating. The finished specimens were annealed again in the hydraulic oil bath at 115 [degrees] C for 4 h to remove the residual stress built into the surface during the machining process.

As demonstrated by Knauss and Kenner (10), the moisture content in amorphous polymers can have a significant effect on the viscoelastic behavior. The volumetric dilatation due to moisture content has the same effect on the creep behavior as temperature if the induced volumetric deformations are the same. To avoid this, the specimens were stored and used at the same relative humidity at all times. An environment of 6% of relative humidity at room temperature was produced via a saturated sodium hydroxide solution within an enclosed belljar (11). The annealed PMMA specimens were stored for two weeks prior to use. The weight of each specimen was measured every few days and found to decrease initially but to remain constant after three days (Mettler electronic balance, model HL 32 with an accuracy of 0.001g). During the measurements the relative humidity was maintained at 6%. Except for the time when the specimen was taken out from the belljar and placed into the environment chamber, the specimen was always in an environment of 6% relative humidity.

Digital image correlation (2) was used to monitor the surface deformation of the specimen. This method requires a uniformly random speckle pattern on the specimen surface, which is attained by first uniformly spraying (Krylon) flat white spray paint on the specimen surface to generate a white background and by subsequently splattering on black paint with the help of a toothbrush.

Physical Test Arrangement

The same MTS test system and Russell environmental chamber were used in the measurements of the uniaxial and the shear relaxation moduli. An image acquisition system consisting of a Nikon F2 camera with a 200 mm objective, a Sanyo CCD camera with 640 x 480 pixel spatial resolution and 8 bits of gray scale, a Data Translation monochrome frame grabber (model DT2855), and a personal computer (486, 66 MHz) were used to automatically acquire digital images at predetermined times during the experiment. The acquired images were processed later using the digital image correlation method (2). An air hole drilled in the lower grip allowed the air inside the hollow cylinder to circulate and equilibrate with the surrounding temperature. The constant axial and shear strains were 0.2% and 0.25%,(6) respectively.


Uniaxial Relaxation Modulus and Poisson's Ratio Measured of the Rohm & Haas PMMA (Plate Specimens)

The relaxation data for this material are shown in Fig. 3 for different temperatures, and the corresponding Poisson data in Fig. 4. Following McLoughlin/Tobolsky (3) and Williams/Landel/Ferry (12) the relaxation data is shifted along the log-time axis to obtain the best-fit master curve in Fig. 5 and the Poisson data shifted by identical amounts in Fig. 6 with the shift factor recorded in Fig. 7. Although there is no common understanding or agreement that such shifting in the glassy region is valid, it has been observed to hold for several materials [e.g., see for PVAc, Emri and Knauss (13), and an epoxy, Matsukawa et al. (14)], though the existence of a pronounced [Beta]-transition might be cause for concern.

The master relaxation curve is fairly smooth in terms of experimental error, however, the master curve for the Poisson data is not. This is due to the error incurred in measuring the extremely small transverse strains (between 0.06% and 0.1%) for a Poisson's ratio between 0.3% and 0.5% for most polymers when only 0.2% of axial strain is applied. It seems also reasonable to log-shift the Poisson data at different temperatures so as to produce a "best fit master curve" independent of the modulus data, as shown in Fig. 8 with the shift factors shown in Fig. 7. The Poisson master curve produced in this way is more narrowly bounded.

The question arises whether the shift factors for the uniaxial modulus and Poisson's ratio data are intrinsically different. Since the error bound for Poisson's ratio is [+ or -]10%, and a smooth master curve can be generated within the error band, the evidence is not sufficient to determine at this time whether the shift factors for uniaxial relaxation and Poisson data are the same or not.

Using this data [ILLUSTRATION FOR FIGURE 8 OMITTED] one obtains the bulk modulus via Eq 1 as recorded in Fig. 9. The error band is small at short times but increases over time. At log(t) = 4, the bulk modulus in the upper bound is 3.67 times of that in the lower bound. The actual bulk modulus might be between the upper and lower bounds. However, these bounds are too large to be able to provide useful bulk data for practical use. Since the actual variation of bulk modulus over a wide range of time (or frequency) has been determined to be small,(7) the results computed from E(t) and v(t) would not be more accurate than a fictitiously constant bulk modulus. This observation casts serious doubt on the validity of computing the bulk modulus from the uniaxial relaxation modulus and Poisson's ratio. A simple quasielastic estimate renders from

[Kappa] = E/3(1 - 2v),

that the relative error in K is, in terms of relative errors in E and v

[absolute value of d[Kappa]/[Kappa]] [less than or equal to] [absolute value of dE/E] + 2v/1 - 2v [absolute value of dv/v]. (8)

For a reasonably good measurement one may expect a 5% of error in E and 10% of error in v. For these error bounds in E and v, the relative error in K is plotted in Fig. 10 as a function of v for 0.3 [less than] v [less than] 0.49, which covers most polymers. This error is much higher than the errors in E and v and increases rapidly as v approaches 0.5; at v = 0.49 it reaches 500%! This large error makes it unrealistic to expect a reasonable determination of K from measurements of E and v, especially at temperatures near the glass transition when Poisson's ratio approaches 0.5. In fact, one readily computes from Eq 8 that for a relative error in the uniaxial modulus and an expected error of 5% for the bulk modulus, Poisson's ratio must be determined with a precision on the order of [10.sup.-]4 as the rubbery plateau is approached (v [approaches to] 0.499), experimentally a more than daunting task.

The "Short-Time (Glassy) Limit" of the Relaxation Modulus

The uniaxial modulus was also determined through ultrasonic measurement at room temperature by determining the wave speeds of a longitudinal and a shear pulse at a frequency of 5 MHz; the uniaxial modulus and the Poisson's ratio were found to be 6.1 GPa and 0.33. By way of comparison, the uniaxial modulus obtained in tension at a strain rate of 0.1%/min was 3.1 GPa. This large difference in the moduli measured at different strain rates suggests that the moduli of PMMA at room temperature (22 [degrees] C) are still rate dependent, rather than reaching a glassy plateau, an observation indicative of the well-established extended [Beta]-transition in PMMA. One may expect thus that the relaxation process will proceed even at temperatures very much below the glass transition. Consequently, the uniaxial relaxation modulus was measured at suitably low temperatures, and the data from -40 [degrees] C to 0 [degrees] C are included in Fig. 3. More data may be found in Lu and Knauss (18). It is clear that the relaxation process is still significant at 145 [degrees] C below the glass transition, contrary to the widely accepted concept that the relaxation is negligible at such low temperatures. See also experiments by Rusch (4).

Uniaxial and Shear Relaxation Data for ACE Material (Tubular Specimens)

Figures 11 and 12 show the uniaxial and shear relaxation moduli at different temperatures for the ACE PMMA. The shift factor in Fig. 13 transforms this data into the master curves [ILLUSTRATION FOR FIGURES 14 AND 15 OMITTED], which are rather smooth in terms of experimental error.

To illustrate the uncertainties in determining the bulk modulus from such measurements it suffices to show this computation for the room temperature data over three decades of time, as shown in Fig. 16. These computations are based on numerically evaluating convolution integrals and illustrate the sensitivity of the bulk behavior to errors in the shear modulus in two ways: Assuming the uniaxial modulus E(t) to be error-free, the measured shear modulus in that Figure, [Mu](t), yields a strongly increasing bulk characteristic. Alternately, assuming a (very) small "correction' to the shear modulus in the form of [[Mu].sub.a](t), as shown in that Figure, brings the relaxation behavior of the bulk function "in line."

As a second way of illustrating the sensitivity of the conversion computation one may use again quasielastic analysis to estimate the required accuracy on the shear modulus function in order to achieve a bulk modulus function within a prescribed range of precision. Using the elastic relation

[Kappa] = [Mu]E/3(3[Mu] - E) (9)

one finds (for dE = 0) that the relative error in the shear modulus is given by

d[Mu]/[Mu] = (1/2 - v) d[Kappa]/[Kappa] as v [approaches] 1/2

Thus when approaching the rubbery domain (v [approaches] 1/2), the shear modulus would have to be determined with about thousand fold precision relative to the desired bulk modulus precision. This requirement is impossible to meet experimentally.

Additional Comparison of Measured and Converted Functions

Let us conclude with two (limited) comparisons, one for the two uniaxial relaxation measurements, and one that inverts this relaxation modulus to the creep compliance for comparison with experimental creep data. The relaxation data compiled from the two PMMA materials (Rohm & Haas and ACE) are juxtaposed in Fig. 17. The agreement is not "perfect" because one deals here with two different sources of material without detailed molecular identification. More important, one should bear in mind that these results derive from measurements at different relative humidities. While no detailed study of the quantitative effect of humidity on the rheological behavior was intended here, one notes that the differences are certainly consistent with the different environments.

In Fig. 18 the shear creep compliance as measured at 22 [degrees] C for the ACE PMMA is compared with the creep compliance for the same material, but as computed from the shear relaxation modulus for the same solid. Within experimental error, the agreement appears acceptable. One only notes that there appears to persist a systematically larger slope for the measured creep behavior, which occurred at an increasing strain larger than that used to determine the relaxation behavior. This (slight) difference is believed to be a vestige of nonlinear behavior, inasmuch as similar comparisons at higher temperatures incur larger creep strains than are typically used in relaxation measurements, and the creep measurements deviate consistently from those computed from the relaxation data by increasingly exhibiting both larger slopes and larger values. Discussion of this type of comparison exceeds the purpose of this presentation and the reader is referred to a more detailed discussion on this topic (19).


Normally accessible viscoelastic material functions have been measured by means of relatively straightforward experimental methods, and it has been illustrated that standard measurement accuracy is totally inadequate to allow conversion of these properties to bulk-related time dependent behavior. Computations indicate and experimental data support the idea that the relative error in bulk data is extremely sensitive to errors in other material functions [E(t), [Mu](t) and v(t)]. Direct measurements over a wide range of temperatures and times (or frequencies) are thus necessary to determine the bulk behavior.


This work was initially supported by the program on Advanced Technologies (PAT) at Caltech under the sponsorship of Aerojet General, General Motors, and TRW and by ONR grant N00014-91-J-1427 with Dr. Peter Schmidt as the monitor. Later work was supported by NASA under grant #NSG 1483, with Dr. Tom Gates as the technical monitor. In addition, funds from the National Science Foundation under grant #CMS9504144 allowed completion of this study.

4 This annealing process was necessary because the as-received material contained residual strains. While there exists a mild anisotropy in the material, that material feature disappears within experimental error after annealing.

5 It was found that if the rod was not annealed before machining, the finished specimen would deform after annealing subsequent to the machining.

6 We mention peripherally that the application of a fixed end rotation of the cylinder produced a steady shear strain in the specimen. This was established with digital image correlation, which yielded very constant strain values with time and in agreement with the value derived from the end rotation.

7 It has been determined by McKinney and Belcher (15) that for PVAc, the dynamic storage compliance decreases by less than 50% over 12 decades of frequency. Recent measurements by Deng and Knauss (16) render a similar ratio while Lin and Nolle (17) determined the ratio as 2.7 (0 [degrees] C [less than] temperature [less than] 550 [degrees] C).


1. P. Heydemann, Acustica, 9, 446 (1959).

2. H. Lu, G. Vendroux, and W. G. Knauss, "Surface Deformation Measurements of a Cylindrical Specimen by Digital Image Correlation," GALCIT SM Report 96-1. California Institute of Technology (1996). Submitted for publication.

3. J. R. McLoughlin and A. V. Tobolsky, J. Colloid Sci., 7, 555 (1952).

4. K. C. Rusch, J. Macrotool, Sci., Phys., B2[2], 179 (1968).

5. I. Gilmour, A. Trainor, and R. N. Haward, J. Polym. Sci., Polym. Phys. Ed., 12, 1939 (1974).

6. I. Wilson, N.M. Ladizesky, and I. M. Ward, J. Mater. Sci., 11, 2177 (1976).

7. I. Wilson, A. Cunningham, and I. M. Ward, J. Mater. Sci., 11, 2181 (1976).

8. D. McCammond and S. Turner, Polym. Eng. Sci., 13, 187 (1973).

9. I. L. Hopkins and R. W. Hamming, J. Appl. Phys., 28, 906 (1957).

10. W. G. Knauss and V. H. Kenner, J. Appl. Phys., 51, 10 (1980).

11. D. R. Lide, Handbook of Chemistry and Physics, 15-25, Chemical Rubber Pub. Co., (1995).

12. M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc., 77, 3701 (1955).

13. I. J. Emri and W. G. Knauss, Comput. Structures, 13, 123 (1981).

14. S. Matsukawa, W. McDonough, and D. L. Hunston, Proc. 15th Annual Meeting of the Adhesion Society, Library of Congress number 92-70209 (1992).

15. J. E. McKinney and H. V. Belcher, J. Res. National Bureau of Standards, A. Phys. and Chem., 67A(1), 43 (1963).

16. T. H. Deng and W. G. Knauss, "The Temperature and Frequency Dependence of the Bulk Compliance of Poly(Vinyl Acetate)," GALCIT SM Report 96-5, California Institute of Technology (1996). To appear in Mech. Time Dep. Mat. (Kluwer Academic Pub., Dordrecht. The Netherlands), 1 (1997).

17. T. S. Lin and A. W. Nolle, Polymer, 30, 648 (1989).

18. H. Lu and W. G. Knauss, "The Stress Relaxation Behavior of PMMA," GAL-CIT SM Report 93-17. California Institute of Technology (1993).

19. H. Lu and W. G. Knauss, "Nonlinear Polymer Response under Blaxial Stress States," GALCIT SM Report 96-2, California Institute of Technology (1996). To appear in Mech. Time Dep. Mat. (Kluwer Academic Pub., Dordrecht, The Netherlands), 1 (1997).
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Title Annotation:International Forum on Polymers - 1996
Author:Lu, H.; Zhang, X.; Knauss, W.G.
Publication:Polymer Engineering and Science
Date:Jun 1, 1997
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