ZHAmtdm07.pdf

Mech Time-Depend Mater (2007) 11: 289–308

DOI 10.1007/s11043-008-9048-7

Applicability of the time–temperature superposition

principle in modeling dynamic response of a polyurea

J. Zhao

W.G. Knauss

G. Ravichandran

Received: 21 September 2007 / Accepted: 28 November 2007 / Published online: 24 January 2008

Abstract

This paper addresses the applicability of the Time–Temperature Superposition

Principle in the dynamic response of a polyurea polymer at high strain rates and differ-

ent temperatures. Careful and extensive measurements in the time domain of the relaxation

behavior and subsequent deduction of a master-relaxation curve establish the mechanical

behavior for quasistatic deformations over a time range of 16 decades. To examine its valid-

ity in a highly dynamic environment, experiments with the aid of a split Hopkinson (Kolsky)

pressure bar are carried out. The use of a two-material pulse shaper allows for stress equilib-

rium across the specimen during the compression process, to concentrate on the initial, small

deformation part that characterizes linearly viscoelastic behavior. This behavior of polyurea

at high strain rates and different temperatures is then investigated by comparing results from

a physically fully three-dimensional (axisymmetric) numerical model, employing the qua-

sistatically obtained properties, with corresponding Hopkinson bar measurements. The ex-

perimentally determined wave history entering the specimen is used as input to the model.

Experimental and simulation results are compared with each other to demonstrate that the

Time–Temperature Superposition Principle can indeed provide the requisite data for high

strain rate loading of viscoelastic solids, at least to the extent that linear viscoelasticity ap-

plies with respect to the polyurea material.

Keywords

Viscoelasticity

Time–temperature superposition

Master curve

High strain

rates

Hopkinson bar

1 Introduction

The behavior of polymers depends strongly on the time-rates of their deformation, on tem-

perature, as well as on (high) pressure (Tschoegl et al.

2002

). This time dependence is un-

derstood to stretch over as many as 20 decades, not all of which can be accessed realistically

J. Zhao

W.G. Knauss (

)

G. Ravichandran

Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA

e-mail: wgk@caltech.edu

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Mech Time-Depend Mater (2007) 11: 289–308

at any one temperature. However, half a century of work involving viscoelastic characteri-

zation of polymers has convinced the polymer physics community through measurements

that a special relation exists between the time scale and the use temperature. In particular, it

has been accepted for a long time that, apart from other small temperature effects related to

rubber elasticity concepts (Treloar

1975

) that, when plotted on a base-10 logarithmic time

basis the physical characteristic response functions—the relaxation modulus or the creep

compliance—translate along the log-time axis by an amount that is a unique function of the

temperature. Generally, “colder temperatures” effect a shifting towards longer times, while

the converse is true for “higher temperatures”. This phenomenon, usually referred to as the

time–temperature shift principle, is well accepted and has been reconfirmed many times for

amorphous polymers at temperatures above the glass transition temperature.

Because the time scales of laboratory tests are typically limited to a few decades

test re-

sults obtained at different temperatures can be used together with this translating or shifting

process to generate response curves that extend over many more decades than the labora-

tory tests would allow

per se

. The result of such a composition process is usually termed a

master curve. An example of this process leading to a master curve that extends over many

decades of time which were not physically encountered in the laboratory is delineated later

in this paper. Because such measurements are universally made in a quasistatic time frame

measured in seconds, minutes and hours, the applicability of the master curve to the much

shorter time frames as encountered in wave propagation problems has been repeatedly called

into question.

One may argue that wave propagation phenomena relate to inertial effects rather than

material behavior as such, so that the same material response characteristics should apply

with equal validity to stress and deformation states generated quasistatically or via inertial

conditions. However, that argumentation has never quite satisfied investigators whose main

efforts concentrate on dynamics problems. As a result, the applicability of data acquired

through quasistatic tests and “extrapolated via the time–temperature superposition process”

to engineering problems involving much shorter time scales has long been in question. It is

the purpose of this publication to address that issue. As a corollary it would seem that if one

can show that the time–temperature reduced data is applicable in dynamic situations for this

particular material, there is a great likelihood—if not a guarantee—that the same should be

true for other amorphous

polymers that obey the time–temperature equivalence principle.

This paper is organized into nine sections, with the remaining ones covering the topics

General comments (Sect.

), Quasistatic relaxation behavior (Sect.

), Dynamic experimen-

tal arrangements (Sect.

), Precision of the experimental method for linearly viscoelastic

response, including pulse-shaping (Sect.

), Computer simulation model, (Sect.

), Com-

parison of experimental and computed dynamic responses (Sect.

), Summary (Sect.

)and

References.

This temperature is a material characteristic that determines the transition between rubber-like behavior and

the (more) rigid state of the polymer.

Sometimes investigators measure polymer creep or relaxation over as short a time scale as 1.5 to 2 decades;

this is not a recommended practice, since 3 to four decades provides considerably more definition for the

mechanical characterization process for constructing the master curve by time–temperature shifting.

This material is actually not a completely amorphous material, since there exists a small amount of crys-

talline domains. However, these relatively rigid phases do not come into play in the temperature domain under

study and should, therefore, not compromise the results.

Mech Time-Depend Mater (2007) 11: 289–308

291

2 General comments

We have limited the investigation to the small deformation range, so that linearly viscoelastic

behavior can be inferred. While many engineering applications are not fully served by this

deformation range, there is a cogent reason for this restriction. Inasmuch as we are interested

in assessing the applicability of properties determined quasistatically by time–temperature

superposition so as to be used in dynamic applications, we must adhere to the rules un-

der which the time–temperature superposition is commonly practiced. Relaxation and creep

properties are consistently determined over a large temperature-reduced time scale for the

small deformation range (linear material response). Because the viscoelastic behavior of all

polymers is poorly understood in the non-linear range, we cannot afford to mix this uncer-

tainty into the examination of whether the temperature-reduced time scales are applicable

in dynamic situations. We can only hope, that if an equivalence can be established for small

deformations, then any extension of research to the nonlinear behavior range(s) may expe-

rience fewer roadblocks.

The interpretation of experimental results is always subject to the quality of data. With

that recognition in mind considerable effort was expended on as careful a material character-

ization as was feasible with the existing equipment. To the extent that equipment improve-

ments could be managed economically, such improvements were incorporated. A serious

challenge arose because of the relatively small size of the (compression) specimens, in ad-

dition to the desire to remain within the linear response realm, so that small deformations

needed to be resolved with considerable precision. In the near-rubbery domain, where low

stiffness dominates, strains on the order of 3.8% or less were deemed sufficient, while at the

higher rates when the material responds more stiffly strains of about 2% or less were consid-

ered adequate. One reason why the higher strains in the rubbery domain were allowed was

that the increased strains also increased the stress level which improved the signal-to-noise

ratio of the load cell signal: loads measured at elevated temperatures tended to be small so

as to fall into the lower measuring range of the load cell.

Finally, existing thermal condi-

tioning facilities were limited to cooling specimens to the glass transition temperature of

polyurea. For the present study, this temperature range was considered sufficient, because

the time–temperature superposition principle rests on its physically and theoretically most

coherent foundation/explanation there.

In determining the relaxation modulus of a viscoelastic material under quasistatic condi-

tions one makes use of simple relations describing homogeneous stress and strain fields in

equilibrium situations such that the modulus property can be extracted readily. For exam-

ple, the determination of the relaxation modulus needs, in principle, “only” the prescription

of a fixed tensile or compressive strain and the measurement of the force (and specimen

cross sectional area) required to maintain that deformation. Such a simple possibility does

not exist with Hopkinson bar measurements, even if the specimen is loaded so that it may

be considered to be temporarily in a state of equilibrium. To illustrate this observation the

reader may, at this point, refer to Fig.

below which demonstrates the strain rate history in

a Hopkinson bar measurement. For the purpose of viscoelastic material characterization it is

thus nearly impossible to prescribe a simple load history in Hopkinson bar equipment (

e.g.

constant strain rate history) since the striker bar contact and pulse shaper impose their own

constraints. In principle one could—and we do—measure the incident force history and the

resulting strain history for deducing the transfer function via an integral equation, the result

A newly purchased, low capacity load cell ultimately proved to lack sufficient precision.

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Mech Time-Depend Mater (2007) 11: 289–308

of which would be the relaxation modulus or the creep compliance. However, for an arbi-

trary, if measured, load or deformation history this property extraction could be tantamount

to a trial and error fitting of a range of viscoelastic functions so as to best match the results

of such correlated measurements.

Because the trial and error fitting could be a daunting task one might argue that much

effort can be saved if one starts out with the quasistatically determined master curve as a

“first guess” and then provide variations around this function to make the Hopkinson bar

measurements correspond with each other. On the other hand, if, under that scenario, the

master curve were to satisfy the dynamically measured correlation between force and de-

formation without further correction, this observation would be equivalent to having shown

that the quasistatically determined master curve does, indeed, represent the material behav-

ior under dynamic conditions. This is the approach we follow in the subsequent work: After

documenting derivation of the relaxation master curve through low-speed test procedures,

we make use of this material function to compute the response of the Hopkinson bar spec-

imen under dynamic conditions, and then compare these computations with the measured

laboratory counterpart.

3 Quasistatic relaxation behavior

Because this study was part of a larger effort devoted to compressive stress fields of elas-

tomers we chose to determine the relaxation behavior under (uniaxial) compression, though

for linearly viscoelastic behavior no distinction arises with regard to the tensile or com-

pressive nature of the imposed deformation. Specimens were used having cross sections of

14 mm

and 30 mm in length.

Preliminary tests on a servo-hydraulic materials testing system (MTS) invariably showed

data irregularities in the start-up phase, though a ramp history with a 10 sec ramp time

instead of one of 1 sec duration improved the recovery of the relaxation modulus. Two par-

adigms were employed to reduce errors and noise effects. Because some of the force tran-

sients were traced to the

gradual

compression and seating between specimen and testing

machine platens, both the specimen and the platens were lapped and polished—the spec-

imen in a special holding fixture—to achieve surfaces that were parallel to within 1 mi-

cron per mm across the specimen diameter. Moreover, to minimize the last vestiges of such

undesirable errors the specimens were pre-compressed through a displacement of 0.02 to

0.04 mm, with a subsequent rest period of at least one hour to allow sufficient time for the

associated relaxation to a (“rubbery”) steady stress before imposing the ramp history (as

the dominant, additive load history). To assure acceptable data consistency commensurate

with the small displacements corresponding to the small strains, the oil in the test frame was

allowed to achieve a constant temperature of at least 40

◦

C during two hours of warm-up of

the MTS system. The temperature control unit was allowed to reach its pre-set temperature

during at least a one-hour start-up period followed by a one hour thermal equilibration of the

specimen. To eliminate any potential thermally induced preload, the relative displacement

between the load-free ends of the test frame were monitored with an LVDT, so that pre-

loading would occur only after that relative displacement was found to be non-detectable.

During testing, the temperature varied within a range of no more than

◦

For additional detail the reader may wish to consult the reference by Knauss and Zhao (

2008

) which deals

with enlarging the time scale over which data can be recovered in a ramp test.

Mech Time-Depend Mater (2007) 11: 289–308

293

Fig. 1

Relaxation behaviors at

different temperatures

Fig. 2

Master curves at 0

◦

Cfor

two specimens generated from

the same material batch (

at 0

◦

Because of the small specimen dimensions and the relatively long specimen support

structure of the test frame which extended into the thermal control chamber, the thermally

induced dimensional changes of this rod connection could result in significant errors in the

strain prescription. For this reason that support was manufactured from Invar (an iron–nickel

possessing very small thermal expansivity), which design feature virtually eliminated that

possible error source.

Figure

records the relaxation data as measured at various temperatures in a ramp strain

history;

these data were then reduced via the standard time temperature shift process, in-

cluding the correction in the modulus for the rubber-elastic response to temperature (modu-

lus

∼

absolute temperature) to produce Fig.

. Figure

incorporates the results of tests on

two geometrically identical specimens, each one of which was, however, cut from a different

Data from both the constant strain rate and constant strain portions were used. The figures refer to the

constant strain rate portion as the “ramp” part and to the constant strain portion as the “post-ramp” history.

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Mech Time-Depend Mater (2007) 11: 289–308

Fig. 3

Shift factor as a function

of temperature

location in the raw material sheet supplied. The first two authors each performed the shift-

ing process separately and independently, with identical shift results as shown in Fig.

.The

WLF equation (Williams et al.

1955

) referenced in the figure was fitted in the most often

used form of

log

−

)

101

−

)

(1)

where the reference temperature

, often valued at 50

◦

C above the glass transition tem-

perature, was taken as unity. This places the glass transition temperature for the polyurea

material at around

−

◦

4 Dynamic experimental arrangements

To access the high strain rate domain we made use of a split Hopkinson (Kolsky) pressure

bar shown schematically in Fig.

. The time scale of these measurements is on the order of

to 10

times shorter than those employed in the quasistatic tests. Split Hopkinson bars

are typically used for non-dispersive materials possessing considerably higher stiffness than

the polyurea material at hand. As a consequence the deformation and stress levels tend to be

rather large and not in keeping within the small deformation goals intended for study here.

For this reason it will be necessary to carefully assess the precision to be expected from this

device at the load and strain levels of interest.

The split Hopkinson pressure bar consists of a striker bar, an incident bar and a trans-

mission bar of 250 mm, 2200 mm, and 1500 mm lengths, respectively; all are of 7075-T6

Aluminum and possess a common diameter of 12.7 mm. Two X-cut quartz gages of 12.7 mm

diameter and 0.254 mm thick were mounted coaxially and close (10 mm) to the specimen

contact surfaces on the incident and transmission bars for recording the longitudinal force

(stress) on both cross sectional specimen surfaces. In addition, 1000

strain gages with a

gage factor of 3.27 were bonded to the incident and transmission bars in the locations shown

in Fig.

to track the strain signals. Because the specimen impedance was typically small

the transmitted signal tended to be weak. Therefore, two strain gages were attached on di-

ametrically opposite sides of the transmission bar, which were then connected as additive

Mech Time-Depend Mater (2007) 11: 289–308

295

Fig. 4

Split Hopkinson bar arrangement

branches in the Wheatstone bridge circuit to double the strain signal and thus to improve the

signal-to-noise ratio.

4.1 Specimen preparation and bar alignment

Cylindrical specimens were used which were typically 3 mm thick and 9 mm in diameter,

though some specimens were cylinders of 6 mm diameter. Special precautions were taken

in preparing these for the Hopkinson bar measurements. A major inaccuracy can derive

from a less than perfect contact at the interfaces between the specimen and the incident and

transmission bars. Of foremost importance is the elimination or at least reduction of an

ini-

tially partial

contact across the end surfaces of the specimen. In traditional Hopkinson bar

measurements this issue is of relatively minor importance since the deformations encoun-

tered are usually so large that perturbations of this kind of magnitude play no significant

role. However, to address the material linearity of the deformation process this geometric

(contact) issue required considerable development effort resulting in the design of a special

lapping fixture that allowed generation both of flat surfaces and their parallel alignment to

within 0.01 mm across the specimen diameter (0.00011 rad

0.064 degrees). To convey the

sensitivity of this alignment of the bar ends relative to the specimen surfaces we note that

this would correspond to a misalignment similar to bending the roughly 2-meter long trans-

mission bar to a central deflection of less than half a millimeter. Thus, to assure near-perfect

alignment of the incident and transmission bars, new and straight bars were specifically

manufactured. A satisfactory alignment was then always guaranteed if, in the absence of

a specimen, the wave transmission across the bar faces produced no more than a “blip” in

the signals, the amplitude of which was small compared to the measurement signals in our

experiments. Such alignment checks were carried out routinely during the sequence of the

measurements reported here. An example of the signals generated with the aid of the com-

posite pulse shaper, discussed later, and in the absence of a specimen is shown in Fig.

.In

this case all traces have been shifted to the same time frame to ease the comparison of the

various signals.

In view of the subsequent modeling analysis thought must be given to the boundary

conditions at the interfaces between the specimen and the incident/transmission bars. To

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Mech Time-Depend Mater (2007) 11: 289–308

Fig. 5

Signals (in the absence of

a specimen) from all gages in the

Hopkinson bar generated with a

composite pulse shaper

consisting of polyurea (1.32 mm

thick, and 8

17 laterally)

and 16 layers of paper

eliminate/reduce the lateral displacement constraint derived from friction at these interfaces

a lubricant was used that operated also satisfactorily at the low test temperatures.

4.2 Temperature control and effect on transducers

To investigate the dynamic response at low temperatures, a small temperature control cham-

ber was constructed from half inch thick PMMA plates to fit into the Hopkinson bar envi-

ronment as shown in Fig.

. The specimen was cooled by allowing the evaporated nitrogen

to flow from the tank directly into the chamber via a control valve. To avoid the purchase or

construction of a thermal control system for the few tests envisioned, it was more resource

efficient to implement manual control. This approach was quite feasible since temperature

control was not required to produce a particularly precise test temperature, since the test

duration was very short. Use of direct nitrogen insertion into the chamber obviated the prob-

lem of introducing (invariably) moist air into the sub-freezing environment with the atten-

dant frosting of the chamber that obscured visual tracking of the set-up inside. To assist in

maintaining a sufficiently stable temperature evolution, four aluminum blocks were located

in the chamber close to the specimen to act as a thermal mass.

The flow valve was controlled manually for a cooling rate of about

−

◦

C/minute. The

human control loop was guided by four thermocouples, of which one was located next to

the incident bar and close to a quartz gage, so as to track the interface temperature, one was

in close proximity to the specimen, and two were attached to a dummy specimen, 2 mm

thick with metal ends, so as to simulate the real specimen, except that the environmental

thermocouples were substituted by one that penetrated into the polymer of the dummy spec-

imen, and the other into one metal end. The latter dummy assembly was also located close

to the test specimen. While the heat flow into and out of the four thermocouples was usually

not coordinated in phase, their temperature differences varied typically by no more than 1

to 3

◦

C. We relied most heavily on the thermocouple embedded in the dummy specimen for

ascertaining the test temperature.

In view of the results discussed in Sect.

it is important to recognize the following

facts: As temperatures approaching the major glass transition, the shift factor changes rather

Low temperature ZP Grease, operational to

−

200

◦

C, and manufactured by MK Impex, Canada.

Mech Time-Depend Mater (2007) 11: 289–308

297

Fig. 6

Temperature control

chamber

(a) Chamber and Hopkinson bars

(b) Specimen inside the control chamber

strongly with temperature and thus an uncertainty of 1 to 3

◦

C induces a more significant

uncertainty in the time dependence of the relaxation modulus. A consequence of this dis-

proportionately large uncertainty is that an “experimentally indicated” temperature may lead

to a computed dynamic response that is not “identical” with the measured one. On the other

hand, if the measured response falls within the range of the computed responses based on the

extremes of the thermal uncertainty band, one may at least conclude that the quasistatically

determined properties do not invalidate their application to dynamic situations.

The question arises as to how the thermal variations affected any transducer outputs. Be-

cause the strain gages were not contained in the temperature control chamber they remained

at room temperature and that issue did not arise, even though these gages were temperature

corrected. The manufacturer of the quartz gages could not provide any reliable data on their

thermal performance so that calibration tests were called for. It turned out, however, that in

the temperature range of interest in this work, no sensitivity to temperature changes could

be detected.

5 Precision of the experimental method for linearly viscoelastic response including

pulse-shaping

It is well known that experiments with Hopkinson bars are normally intended to access high

strain rates and large deformations. However, because of our interest in linearly viscoelastic

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Mech Time-Depend Mater (2007) 11: 289–308

response and correspondingly small response signals, the performance of this experimental

tool needs to be carefully assessed and controlled.

Before showing the results of this process we comment first on the way in which wave

propagation data is presented here: Because the strain and quartz gages are located at dif-

ferent stations along the length of the Hopkinson bar the time histories of their signals,

specifically of the wave arrival times, will be time-spaced according to their locations. How-

ever, to facilitate signal comparison we display the signals for the incident bar time-shifted

so as to refer each one to its respective arrival time; they appear, therefore, as if they were to

originate at the same time. The same applies to the signals from the transmission bar, though

the time differential between the incident and the transmission bars is maintained. This op-

eration can be performed with some precision because the distances between the gages and

the attendant wave propagation speeds are well known.

The equations typically used for Hopkinson bar studies require that the time history be

such that the specimen is essentially in a homogeneous state of stress and deformation. In

contrast to when wave motion dominates

within the specimen

, we refer to this state also as

“a dynamic equilibrium state” or simply “equilibrium state”. Even if that condition is con-

ceptually not an absolute requirement for our study—in view of the fact that comparison

with computational modeling is involved—its use simplifies the analysis and interpretation

of the dynamic signals markedly, if for no other reason than to “trouble-shoot” the measure-

ments. With the relatively high importance of precision measurements required in this study,

we found this latter aspect very important. To achieve specimen equilibrium one typically

controls the shape of the incident wave by “pulse shaping” through the insertion of a suitably

configured material element between the striker and incident bars. We are well aware that

this pulse shaping comes at the expense of somewhat reduced strain rates, accounting for al-

most a decade in the short-time frame, yet considered this sacrifice worth while to guarantee

the requisite precision of the measurements.

Under the equilibrium conditions the pertinent equations are, with the subscript defini-

tions

incident,

reflected

transmitted, and

referring to the specimen,

∫

(ε

−

)dτ

=−

∫

(ε

)dτ

(2)

(ε

−

)

=−

(3)

(ε

)

(4)

where we remind the reader that elastic relations apply since the quantities refer all to mea-

surements deduced from the strain and quartz gages attached to the elastic aluminum bars.

The desire or need to deal with a specimen in equilibrium generates a significant chal-

lenge that derives from the large impedance mismatch between the specimen and the alu-

minum bars. To deal with a similar situation involving materials of relatively low rigidity,

Chen et al. (

2002

) introduced the pulse shaping technique, which seems to work quite well

for polymeric materials under large deformations. For the present purposes we have ex-

plored pulse shapers of different materials, for example copper, various polymers and wood,

but obtained unsatisfactory results for reaching stress equilibrium when loading forces were

smaller than about a hundred Newtons. Chen et al. (

2002

) encountered the same problems

but were able to ignore these issues because they were minor perturbations on the larger de-

formation results that were of interest to them. Through a trial-and-error process we found

that employing a composite pulse shaper consisting of a combination of polyurea (1 to 2 mm

Mech Time-Depend Mater (2007) 11: 289–308

299

Fig. 7

Room temperature record

of forces at the quartz gages for a

pulse shaper of polyurea only

thick) and 16 sheets of paper towel

in tandem provided a highly acceptable shape with a

minimal sacrifice of short time/high rate performance.

Figure

shows the force histories deduced from the wire-strain and quartz gages, when

only a polymer pulse shaper was employed. Here the notation “incident strain” in the caption

of that figure signifies that the force was deduced from the incident and reflective waves (

both of which are recorded by the incident strain gage. The “transmission strain” in Fig.

denotes that the force is deduced from the transmission wave recorded by the transmission

strain gage. Also, Q1 and Q2 refer to the force histories recorded by the incident and the

transmission quartz gages, respectively. We see that the force histories recorded by the in-

cident strain gage and the incident quartz gage are consistent with each other except for

the oscillation in the quartz gage signal, but they are not consistent with the force histories

recorded by the transmission gages. However, when the composite pulse shaper is employed

as illustrated in Fig.

, one arrives at a very consistent agreement.

The result of this pulse shaping is illustrated in Figs.

and

: both figures demonstrate

the repeatability of separate test runs, with Fig.

indicating that a small temperature dif-

ference near

−

◦

C does not produce a measurable difference in the gage readings. By

contrast, Fig.

demonstrates that significant temperature differences result in clearly dis-

tinguishable gage records. For completeness of presentation we show in these two figures

extended time records which involve, however, also relatively large deformations of no di-

rect interest in this study which, as rationalized in Sect.

, is geared to small deformations.

The reader should bear in mind that linearly viscoelastic behavior for this material is rea-

sonablyassuredifstresslevelsarelimitedto5to6MPaortoaforcelevelontheorderof

400 N

which is the range exhibited in Fig.

. Theses results demonstrate that much of the

total test history typically involves relatively large stresses and strains exceeding the linear

domain and that a comparison with linearly viscoelastic computations is expected to deviate

progressively as nonlinear material behavior enters.

To apprise the reader of the special care in data evaluation required in the present case

we comment on the special way in which data evaluation had to occur, which result from

Georgia Pacific, Roll towel, part # 26100.

Maximal strains on the order of 3.5% at room temperature, smaller at lower temperatures, typically

around 2%.

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Mech Time-Depend Mater (2007) 11: 289–308

Fig. 8

Room temperature record

of forces at the quartz gages for a

pulse shaper of polyurea plus

paper (pulse-shaper: polyurea

8mm

16 layers of

tissue paper cylindrical specimen

9.5 mm

3mm)

Fig. 9

Extended time history of

quartz gage records on the

incident bar (

upper

)and

transmission bar (

lower

)fortwo

specimens

the small deformation limitation in combination with the particular pulse shaping technique

and the available equipment. These factors introduce a limitation not usually encountered

in experiments involving larger deformations and better impedance matching. In the latter

cases (

)–(

) usually allow the determination of the specimen strain from either the reflected

signal or from the difference between the incident and transmitted signal. This alternative

approach is not valid here. Because the striker bar does not rapidly rebound from the incident

bar as a result of the special pulse shaper, the signal in the incident bar is not as “clean” as

if such a separation had occurred on impact: it contains reflections within the striker bar,

that are troublesome to isolate definitively for analysis purposes; consequently the reflected

signal has not been used in this analysis.

Figure

demonstrates this by displaying the signal from the strain gage on the incident

bar and that in the transmission bar. The part of the signal marked as “weak disturbance”

is the low amplitude oscillation in the strain gage record due to the superposition of the

less-than-perfect signal from the striker bar with the reflection at the incident-bar/specimen

interface. The amplitude of this “disturbance signal” is small compared to the peak of the

transmitted signal. However, when the strain in the specimen is derived from the signal

Mech Time-Depend Mater (2007) 11: 289–308

301

Fig. 10

Extended time history of

quartz gage records on the

incident bar (

upper

)and

transmission bar (

lower

)forthe

same specimen at three

temperatures

Fig. 11

Identification of a weak

disturbance arising from the

dispersion of the signal at the

incident bar–specimen interface

reflected at the incident specimen surface (see (

)) the “disturbance” amounts to a major

component of the reflection signal. On the other hand, if one deals only with the—relatively

small—transmission signal, that signal is much less encumbered by the “disturbance” and,

as a consequence, yields a well defined measure of the specimen strain.

To illustrate this observation we anticipate results addressed in the next section and com-

pare the computations of stress and strain histories in the specimen for the two ways of

determining the specimen strain. Figure

a shows the strain history in the specimen as

determined from the reflected signal and from the difference between the incident and trans-

mitted ones. The latter provides a much closer fit to the computed history, which is in close

agreement with that derived from the incident/transmission combination. An identical result

is shown for the stress history in Fig.

b. Based on these results we adhere in the sequel to

determining the specimen deformation and stress histories via the incident and transmitted

signals.

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Mech Time-Depend Mater (2007) 11: 289–308

=−

∫

(ε

)dτ

;b:

∫

(ε

−

)dτ

;

c: simulation

(a) Strain

(ε

)

;b:

;

c: simulation

(b) Stress

Fig. 12

Improvement of data reduction through elimination of the reflected strain in favor of the incident

and transmitted strains. Specimen and pulse shaping are the same as those in Fig.

6 Computer simulation model

As indicated in Sect.

the assessment of how well the quasistatically derived master curve

represents the material response under dynamic conditions is best accomplished with the aid

of a computational or simulation model. We have used the LS-Dyna 970 code which permits

linearly viscoelastic material characterization in the form of relaxation or creep functions in

Mech Time-Depend Mater (2007) 11: 289–308

303

shear

on the one hand, and either a constant bulk modulus or a constant Poisson ratio, on

the other. Because the bulk modulus typically varies by a much smaller amount over the

viscoelastic transition range than the shear or uniaxial modulus it is a very good engineering

approximation to represent the volumetric response by a time-independent bulk modulus.

We note in passing that this approximation is more realistic in the transition range and still

shorter times than choosing a constant Poisson characterization, because the latter choice

links the bulk and shear (or uniaxial) modulus by a direct proportionality, a situation that

is not generally supportable as the glassy range is approached. We choose as the best ap-

proximation the time-dependent, uniaxial (Young’s) modulus and the corresponding shift

factor as summarized above, and a constant bulk modulus of 263 GPa

(Chakkarapani et al.

2006

). It turns out in the present case, however, that this choice of the bulk modulus also

happens to allow the use of a constant Poisson’s ratio in the time and temperature range of

interest.

To guarantee that the code was able to reliably represent complex time histories of non-

trivial initial/boundary value problems we compared its output to analytically tractable and

closed-form solutions for viscoelastic problems using relatively simple material models (one

or two relaxation times). Without recounting here all the details of these analyses, we simply

report in the interest of brevity that in all test situations the code performed very well and in

accordance with the anticipated results.

For the modeling process the Hopkinson bar test configuration was reproduced dimen-

sionally except for the length of the incident and transmission bars: Because of the axisym-

metric geometry the problems was formulated mathematically as two-dimensional, includ-

ing the discontinuities at the transitions from the specimen to the adjacent bars. To allow

for possible signals from these geometric changes the specimen discretization employed 15

uniformly spaced (ring) elements in the radial and 10 elements in the axial direction. The

incident and transmission bars were modeled as 1500 and 1000 mm in length, respectively,

each one divided into 12 coaxial rings of equal thickness and 0.53 mm in length, distributed

uniformly along their lengths. The specimen and bar discretization was implemented via

(axisymmetric) four-noded elements for a total element count of 60150, 150 of which are

allocated to the specimen.

As input into the specimen we used the experimentally determined output of the strain

gage on the incident bar. The properties for the aluminum bars used were: Young’s modulus

(7075-T6 aluminum)

72 GPa, Poisson’s ratio

0.33 and the density of 2.785 g/cm

.The

constitutive equation of the polyurea polymer was chosen in the form of a Prony series for

the shear relaxation modulus as

G(ξ )

∞

∑

−

ξ/λ

(5)

The code does not allow for the use of the uniaxial modulus. This was the function available for the

polyurea, however. For this reason the shear modulus was derived from the unaxial modulus by the approx-

imation valid in the rubbery domain, that

μ(t)

E(t)/

3. In the transition domain and at shorter times this

approximation becomes less accurate; however, for the measured bulk modulus the deviation from reality is

within the range of the experimental error.

It is of parenthetical interest to note that the value of the bulk modulus is not very important, as long as it

is large. Computations with values of 263 GPa on the one hand, and 2 GPa on the other rendered results that

were indistinguishable within plotting accuracy.

304

Mech Time-Depend Mater (2007) 11: 289–308

Fig. 13

Simulation model (only

specimen-near portions of

incident and transmission bars

are shown; one half of

axisymmetric configuration)

Fig. 14

Experimentally

determined strain rate for

conventional polyurea specimen

(6 mm dia

2.65 mm) under

Hopkinson bar loading, as

determined from

(

C/l

)(ε

−

)

where

specimen length

and

are the incident and transmitted

strain signals, respectively

For current purposes it suffices to let

be on the order of six with

suitably chosen to

represent the time scale of the anticipated deformation and stress relaxation. For this limited

time scale a representation with all relaxation times would yield no different results.

7 Comparison of measured and computed dynamic responses

Using the combined pulse shaper technique described above, polyurea specimens 3 mm

thick and 9 mm in diameter were generally employed to explore their viscoelastic response.

Figure

shows the strain rate in the compression process under Hopkinson bar loading,

which varies continuously from near zero to around 10

/s before the strain reaches 4% com-

pression. Higher strain rates are achieved at higher strains, but these take us out of the range

of small-strain viscoelastic behavior. It was thus not possible to deduce the relaxation be-

havior from the Hopkinson bar data through a closed form solution. To follow a completely

numerical-mathematical route which is possible, is, however, equivalent to the procedure

we follow subsequently because we know the relaxation modulus from quasistatic mea-

surements with the aid of the time–temperature super-position principle. If we assume for

computational purposes that these data apply to the dynamic situation we can test that as-

sumption by comparing the computations with the experimental results.

Figure

shows a comparison of the force recorded via the quartz gages and the simula-

tion at the location of the quartz gages in the bars. The excellent agreement in the measured

and computed force histories indicates that we can compute the counterparts to the dynamic

Mech Time-Depend Mater (2007) 11: 289–308

305

Fig. 15

Comparison between

dynamic experimental (quartz

gage) and simulation results

Fig. 16

Comparison between

the experimental and simulated

stress–strain relation under

dynamic loading

measurements. A further confirmation of this recognition is the comparison of the computed

and measured force in Fig.

which is associated with the strain rate history in Fig.

We reiterate that the test for the applicability of the time–temperature shifted, quasistati-

cally obtained master curve is the degree to which this quasistatic data reproduces computa-

tionally the measurements in dynamic situations. Because there were limitations to obtaining

quasistatic measurements at and above room temperature as well as below the glass transi-

tion (see Sect.

), the time range accessed by the Hopkinson bar tests is correspondingly lim-

ited. We have, therefore performed comparison of dynamically recorded stress responses at

five nominally different temperatures (room temperature, 0

◦

−

◦

−

◦

Cand

−

◦

with the lowest two calling for some special discussion.

Regarding the tests at sub-room temperature conditions we start with the dynamic tests

at 0

◦

C. Figure

shows again the comparison of the recorded data and the corresponding

computed responses at the two quartz gages (Q1 is located on the incident bar and Q2 on

the transmission bar). The agreement is very satisfactory, indicating that the data acquired

306

Mech Time-Depend Mater (2007) 11: 289–308

Fig. 17

Comparison between

experimental (quartz gage) and

simulation results at 0

◦

Fig. 18

Comparison between

measurements (quartz gages) and

simulation results at

−

◦

quasistatically in the time frame of 10 to 10

seconds applies to the dynamic problem which

proceeds in the time range on the order of 10

−

seconds (c.f. Fig.

recorded at room tem-

perature).

Because there was relatively little change between the response at 0 and

−

◦

C, we

turn to the dynamic tests conducted at

−

◦

C, illustrated in Fig.

. We note again that the

comparison between computed and measured values at the two quartz gage locations are

very favorable. A repeat of the same tests at nominally

−

◦

C yields the results in Fig.

The comparison of the physical measurements with the computational model at (nomi-

nally)

−

◦

C, and especially at the nominal

−

◦

C temperature requires some additional

discussion. It will be recalled that the thermal control for the split Hopkinson bar (Fig.

)

provided thermal specification and determination only within a small range of temperatures,

which range was estimated to be on the order of 1–3

◦

C. From Sect.

4.2

we note that the

relaxation modulus becomes very sensitive to temperate changes at these low temperatures.

As a result of these constraints it turns out that the computations for (nominally)

−

◦

Cdo