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1988
An Experimental characterization of the tensilebehavior of polystyreneLouis T. Loggi
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Recommended CitationLoggi, Louis T., "An Experimental characterization of the tensile behavior of polystyrene" (1988). Thesis. Rochester Institute ofTechnology. Accessed from
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An Experimental Characterizationof the
Tensile Behavior of Polystyrene
byLouis T. Loggi
A Thesis Submittedin
Partial Fulfillmentof the
Requirements for the Degree ofMASTER OF SCIENCE
inMechanical Engineering
Prof.------------(Thes}s ~dvisor)
Prof. Alan X. Nige
Prof.------------
Prof. P Marletcar(Department H~ad)
DEPARTMENT OF MECHANICAL ENGINEERINGCOLLEGE OF ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGYROCHESTER, NEW YORK
APRIL 1988
THESIS TITLE:
AN EXPERIMENTAL CHARACTERIZATION
OF THETENSILE BEHAVIOR OF POLYSTYRENE
I, LOUIS T. LOGGI, HEREBY GRANT PERMISSION TO THE
WALLACE MEMORIAL LIBRARY OF R.I.T. TO REPRODUCEMY THESIS IN WHOLE OR IN PART. ANY
REPRODUCTION WILL NOT BE FORCOMMERCIAL USE OR PROFIT.
DATE: April 22, 1988
TABLE OF CONTENTS
Page
LIST OF FIGURES i i i
LIST OF TABLES v
ABSTRACT vi
1.0 INTRODUCTION 1
2.0 LITERATURE SEARCH 4
3.0 EXPERIMENTAL TEST PLAN 9
3.1 General Information 9
3 . 2 Test Parameters 11
3.3 Data Analysis 13
4.0 EXPERIMENTAL RESULTS 25
5.0 VISCOELASTIC THEORY 34
5.1 Maxwell Model 34
5.2 Kelvin Model 39
6.0 BRINSON MODEL 43
6.1 Description 43
6.2 Computer Adaptation of Model 44
6.3 Comparison to Experimental Results 46
6.4 Conclusions 48
7.0 MATSOUKA MODEL 65
7.1 Description 65
7.2 Computer Adaptation of Model 68
7.3 Comparison to Experimental Results 70
7.4 Conclusions 72
8.0 CONCLUSIONS 82
REFERENCES 83
APPENDIX A. DATA LOGGER ACQUISITION PROGRAM 85
APPENDIX B. BRINSON MODEL PROGRAM 91
APPENDIX C. MATSOUKA MODEL PROGRAM 97
LIST OF FIGURES
1. Stress-Strain Diagram for Polymers
2. Instron Universal Test Machine and Data Acquisition Equip. (Diagram)
3. Instron Universal Test Machine and Test Specimen Set-up (Photo)
4. Instron Control Console (Photo)
5. Instron Programming Panel & Data Logger Rear Panel
6. Extensometer on Sample (Photo)
7. Strain Gage on Tensile Test Sample
8. Vishay VE-20 Digital Strain Indicator (Photo)
9. Comparison of Extensometer Test Results for 5mm/minute Test Speed
10." "
50mm/minute Test Speed
11."
"lOOmm/minute Test Speed
12. Extensometer Test Results at Four Different Test Speeds
13. Comparison of Strain Gage Test Results
14." "
and Extensometer Results
15. Maxwell Viscoelastic Model
16. Creep Behavior of Maxwell Model
17. Stress Relaxation Behavior of Maxwell Model
18. Kelvin-Voight Viscoelastic Model
19. Delayed Elasticity Behavior of Kelvin Model
20. Incomplete Relaxation Behavior of Kelvin Model
21. Brinson's PMMA Data as a Function of Strain Rate
22. Extensometer Test Results at Slow Crosshead Speeds
(0.5 to lOmm/minute), 0-16% Strain
m
LIST OF FIGURES (Cont'd)
23. Extensometer Test Results at Slow Crosshead Speeds
(1 to 10 mm/minute), 0-5% Strain
24. Extensometer Test Results at Faster Crosshead Speeds
(10 to 500 mm/minute), 0-5% Strain
25. Brinson Model Best Fit Results vs Experimental Data at 1 mm/minute
26. Brinson Model Best Fit Results vs Experimental Data at 500 mm/minute
27. Brinson Model Predicted Response for 5 mm/minute Test Speed
28." "
10 mm/minute Test Speed
29. Brinson Model Best Fit Results vs Experimental Data at 100 mm/minute
30. Brinson Model, Second Predicted Response for 5 mm/minute
31. Relaxation Modulus Comparison, Maxwell vs William Watts Viscoelastic
Models
32. Matsouka Model Best Fit Results vs Experimental Data at lmm/minute
33." "
lOOmm/minute
34. Matsouka Model, Predicted Response for lOmm/minute Test Speed
35. Matsouka Model w/Beta Zero Exponent, Best Fit Results vs Experimental
Data at lOOmm/minute
iv
LIST OF TABLES
1. Detailed Flow Chart of Data Acquisition Progress
2. Test Speeds (mm/minute) and Corresponding Strain Rates
3. Common Viscoelastic Models
4. Brinson Model Data for 1 mm/minute, Final Fit
5. Brinson Model Data for 500 mm/minute, Final Fit
6. Brinson Model Data for 5 mm/minute, Predicted
7. Brinson Model Data for 10 mm/minute, Predicted
8. Brinson Model Data for 5 mm/minute, Second Prediction
9. Brinson Model Summary, Tau vs Strain Rate
10. Matsouka Model Data for lmm/minute, Final Fit
11. Matsouka Model Data for lOOmm/minute, Final Fit
12. Matsouka Model Data for lOmm/minute, Predicted
13. Matsouka Model w/Beta Zero Exponent, Data for lOOmm/minute, Final Fit
ABSTRACT
The characterization of polymeric materials involves many different
tests which often depend upon the ultimate environment for the end
product. One important area for many engineering applications is that
of mechanical testing.
Although the evaluation of tensile properties ranks as the most
common mechanical test, standard test methods can have serious
shortcomings when applied to plastics.
This work seeks to study test method variability by evaluating
experimental data from uni -axial tension tests conducted on a typical
high-impact polystyrene at room temperature. Two common methods will be
used to obtain the strain data: extensometer transducer and foil strain
gage. These results will be compared for repeatability and accuracy.
In addition, the experimental data will be used to determine if a
simple prediction of the viscoelastic mechanical behavior can be made.
The objective here is not to compare the multitude of theories regarding
polymer behavior, but rather, to define some applicable theories which
could be easily supported by experiments.
vi
1.0 INTRODUCTION
The need to describe the properties of a material leads to a
variety of mathematical relationships; equations of state, kinematic/
dynamic principles, and constitutive equations. However, it is the
constitutive equations which help define the particular nature of the
material .
For example, with the assumption of small strains, metals at normal
temperatures behave in a well-ordered fashion. Thus, the constitutive
relations, otherwise known as Hooke's Law, are simple and linear.
In one-dimension, Hooke's Law can be written as:
or = Ee (1-1)
where or = stress, units of force/area such aslb/in^
= strain, dimensionless
E = proportionality constant;
Young's (Elastic) Modulus, units of force/area
Similarly, viscous fluid behavior can be described by Newton's Law:
o-
-1& d-2)
where & = stress
= strain
z = first derivative with respect to time
n = proportionality constant; viscosity
From a test standpoint, relationships of this nature lead to
standardized procedures and repeatable end results. In addition, by
understanding the basic deformation process, it is easier to evaluate
the effects of various test conditions on the material and its behavior
in service (*).
The behavior of polymers, however, can vary considerably from the
predictions of Hooke's Law, even at room temperature. In addition, the
material properties of plastics can change due to sample preparation
and environmental conditions. All of these factors impact typical
design parameters such as the elastic modulus defined above.
Just one example of the difficulty of extending Hooke's Law to
plastics concerns the definitions of the yield and ultimate conditions.
For polymers, yield is defined as the point of maximum tensile strength,
or, in some instances, as the first inflection point of the stress-
strain curve. A 0.2% offset yield criteria generally used for metals
has no physical significance for a polymeric system. Similarly, the
ultimate tensile strength for plastics corresponds to the stress at the
fracture point. (Figure 1) (2)
Therefore, repeatability of experimental results may be inconsistent
at best. This leads to a desire for more specific test conditions and
perhaps a theoretical model to describe the deformation. Both of these
items would hopefully reduce the number of mechanical tests required to
characterize a specific polymer.
TCHWLC tTROMTH AT BREAK
CLOMOATION AT
TE**LE tTRWTH AT TLP>
CLOMOATtON AT ViELO
TEWBtLE BTREBS AT BREAK
ELONBATIOM AT BREAK
TENBK.E BTREBS AT YCLO
ELONBATION AT YIELD
STRAW
Stress 3.:.. Diaqrair, Jor 1-oiymers
Figure 1
3
2.0 LITERATURE SEARCH
The basic intention of this work was to quantify the predominant
mechanical test method and implement a predictive math model, to be used
in lieu of additional testing.
This premise necessitated a two-pronged literature investigation.
One objective was to determine what experimental results had already
emerged from uni -axial tensile tests. In addition, various theoretical
viscoelastic models were examined and judged by the following criteria:
(I) applicability to uni-axial testing
(II) conditions required to obtain experimental data
(III) complexity of the mathematical model
With these ideas, engineering abstracts for the years 1970-1983
were manually researched, with the results compiled as follows:
2.1 EXPERIMENTS
In this area, a wide diversity of data was found. G'sell and Jonas
(3) developed a diameter transducer in order to evaluate the local true
strain rate during uni-axial testing. However, their constitutive
equations were based on strain hardening relationships and were
restricted to non-standard circular test specimens.
Gilmour (4) reviewed existing literature values for elastic
moduli. He also provided experimental results from three test methods:
(a) uni-axial tension using samples of different lengths
(b) uni-axial tension using strain gages
(c) compression of cylindrical samples
All of the data was presented in statistical and tabular form. While no
specific mathematical relationships were given, the results provided a
potential source for comparison.
Yokouchi (^) examined the dependency of polystyrene's mechanical
properties on tensile test speed. A portion of the work utilized a
conventional tensile tester to determine elastic modulus and yield
stress. Reported results included a dramatic increase of maximum load
at test rates greater than 5 m/sec, and a correlation between stress
whitening and the strain rate at failure. Even though no theoretical
equations were presented, this paper served to validate the choice of
uni-axial testing.
A large amount of work was confined to areas beyond the scope of
this thesis. Many of these were studied for general representation and
applicability of experimental data. (6> 7 8)
2.2 THEORIES
An early theoretical model for viscoelastic behavior was developed
by Naghdi and Murch (9). This model is based upon the assumption that
the viscoelastic strain rate is characterized by the creep function
defined for shear loading. The constitutive equations for both
viscoelastic and plastic behavior are shown to depend upon both the
time-stress history and the path or loading surface time history. These
complex relationships included the effects of work hardening and the
state of loading or unloading of the material; items which do not lend
themselves to experimental verification. However, this work has served
as a basis for additional theoretical development.
Schapery UO) devised other linear and non-linear constitutive
equations for uni-axial loading conditions. The main thrust of this
work was to use actual creep experimental data to evaluate various
material property functions. The theory which evolved was confined to
creep and stress relaxation tests, thereby creating a need for this data
for various polymers. Uni-axial tension was the important consideration
for this thesis due to the ease and timeliness of acquiring data. Thus,
this theory was not considered here for validation testing.
An entirely different approach was presented by Bodner 01). He
considered both elastic and plastic deformation components to be present
at all times. Thus, no assumptions about the yield condition are
necessary, and the stress tensor is analogous to the response of a
simple Kelvin-Voight viscoelastic model. While this theory adequately
describes the behavior of metals at elevated temperatures, its
prediction of a constant yield strength is not representative of most
amorphous polymers.
For many concepts, isothermal conditions are a necessary
assumption. However, Rubin (12) included the rate of temperature change
in his model and emerged with a rate-dependent yield strength.
Specifically, the non-linear constitutive equations assume a yield
function which depends upon the total strain rate and the temperature
rate. These relationships involved numerous stress and thermodynamic
functions which would have been difficult to model and validate with
experimental data.
Many theoretical papers discussed very specific topics such as the
relationships between molecular structure and deformation behavior while
others concentrated on alternative mathematical approaches for modeling
the mechanical behavior. These were rejected as being too restricted
for this thesis. (13> 14)
One experiment based model emerged as having the most promise with
respect to the aforementioned criteria. H.F. Brinson (*5) developed a
viscoelastic model in which he defined three regions of polymer
behavior: elastic, viscoelastic, and plastic. The distinction here is
that the elastic limit is defined as the point at which linear visco
elastic behavior begins, where this limits occurs below the yield point
of the material. This model will be discussed in detail in Chapter 6.0
Another promising mathematical model developed from a theoretical
viewpoint. S. Matsouka (16) updated the linear viscoelastic Maxwell
model allowing for a distribution of relaxation times instead of a
single one. In addition, he accounts for differences in strain level
and developed a predictive model which utilizes empirical constants.
This model will be discussed in detail in Chapter 7.0.
3.0 EXPERIMENTAL TEST PLAN
3.1 General
The experiments for this study utilized standard lab procedures
and techniques in order to avoid situations which could not be easily
duplicated in an industrial environment.
Test samples were injection molded polystyrene tensile bars, Type
M-I of ASTM Specification D638M. These are the common"dogbone"
shape
samples, approximately 200 mm x 20 mm, with a narrow section width of
about 12 mm.
All experiments were conducted on an Instron Model 1125 Universal
Test Machine. This particular machine was configured in metric units;
thus, all data is presented in those terms.
To accomplish the data acquisition, the Instron was connected to an
HP 3947A data logger which was controlled by an HP series 200 mini
computer. Interactive software was written to coordinate the task.
After clamping the test sample in the machine, the operator must
manually zero the data logger readings, to avoid any bias errors in the
data. The computer program is then started, which prompts the operator
to enter specific test parameters such as test speed, load cell range,
and type of transducer.
Comments instruct the operator to verify the inputs to the data
logger. Load information is provided from a specific pin location on
the Instron console control programming panel. Strain transducer output
can come from either the programming panel or from a remote device, as
is described below. Details of the test set-up are found in Figures 2
through 5 and Table 1. To begin the test, the operator must start the
Instron manually, as it has no servo motor and thus is incapable of
being computer controlled. A single keystroke on the Series 200
computer will then acquire all load and strain data in voltage form and
convert it to engineering units. The operator is given the option to
reject the data or store it on magnetic floppy disk. A complete listing
of the acquisition program is included in Appendix A.
10
3.2 Parameters
For uni-axial tension testing of a given material, two critical
parameters contribute significantly to the overall variability of the
results:
(a) test speed
(b) type of transducer
Test speeds available on the Instron range from 0.1 mm/minute to
500 mm/minute. ASTM specification D638M for tensile testing recommends
rates of 0.2, 2.0 or 20 mm/minute, depending upon the material
specification or the elapsed time until rupture; however, different test
speeds may be more indicative of the polymer's ultimate environment.
Higher test speeds might simulate impact conditions while lower speeds
may approximate fastening operations.
For this analysis, seven test rates were selected, over a range
from 0.5 mm/minute to 500 mm/minute (see Table 2).
Transducer options included a strain gage extensometer, a uni-axial
strain gage, and measurement of the test machine crosshead displacement.
11
The overall crosshead displacement is highly dependent upon the
type of clamping and the applied preload on the sample. In addition,
the gage length for subsequent strain calculations is not constant from
test to test and is difficult to measure accurately. All of these items
result in poor repeatability.
To compound these difficulties, it is not possible to obtain a
voltage signal from the Instron which is directly related to crosshead
displacement. The displacement indicated on the control console results
from decoding a pulse train from an optical encoder; a task beyond the
scope of this project. Some attempts were made to read load and strain
values from a strip chart recorder, but this method was not of the same
order of accuracy as the other transducer measurements. As such,
crosshead displacement was eliminated from consideration.
A strain gage extensometer is a Wheatstone bridge device which
measures displacement by riding on the test sample (Figure 6). On the
Instron machine, the extensometer has knife edges to avoid large contact
area with the sample, with the remainder of the electronics housed in
the control console. This enables the operator to easily balance the
bridge output after the extensometer is installed. For this study, an
Instron Model 2630-008 extensometer was used, with a gage length of
50 mm and a maximum strain capability of 50%.
12
Similarly, a single foil strain gage will provide accurate results
of the sample's tensile deflection. For these tests, a general purpose
gage was selected; Micro-Measurements Model EA-06-250 BG gage with
120 ohm resistance, a gage length of 6.35mm, and a maximum strain
capability of approximately 5% (Figure 7). To acquire the data, a
Vishay Ellis VE-20A digital strain indicator was utilized (Figure 8). A
half-bridge set-up with a temperature compensation gage was used, with
the output of the VE-20A relayed to the data logger.
3.3 Data Analysis
The actual experimental data for each test run consists of
tabulated results of stress versus strain. It must be noted that all of
the values represent engineering stress and engineering strain.
Measurement of true stress and true strain requires monitoring the
change in specimen size during the test. This is a difficult task which
is not commonly employed in an industrial environment where the thrust
of this work is directed.
To quantify the variation of results that emerge during polymer
tensile testing, each test condition was repeated a minimum of three
times. In this fashion, it was hoped that some statistical limits could
be defined.
13
Analysis consisted of evaluating data from different test runs for
accuracy and repeatability. In addition, experimental results were
compared to computer simulation output from both the Brinson and
Matsouka viscoelastic models. Data format for this comparison is
presented in both graphical and tabular modes.
To quantify these results, a least squares analysis was conducted.
A multiple regression analysis was not done; but rather, experimental
data was used to determine some of the parameters in each viscoelastic
model. The least squares analysis was then conducted only on the single
remaining variable.
Two calculations were used as a measure of the fit, root mean
square error and normalized error. Root mean square error is expressed
in the form:
rms error =
il (data
- estimate)2 (3-1)
N-2
where N = the number of data points.
In order to obtain an idea of the error with respect to strain, a
normalized error term was calculated:
Norm error =
?data - estimated2
(3-2)E
strain
14
TABLE 1
Data Acquisition Flowchart
I. Connect cables from Instron console programming panel to data
logger.
II. Calibrate Load Channel
1. Instron load cell amplifier settings
a. Load Filter'Out'
b. Polarity = 1 (indicates tension)
c. Outer load range dial set = 250
2. Close channel 16 on data logger
3. Press and hold'zero'
button on Instron load cell amplifier
a. Adjust pot until data logger reads 0.0 volts
b. Release button
4. Set inner range dial = 5 (lowest range in white portion of
dial)
a. Adjust balance pot until data logger reads 0.0 volts
b. Move inner range dial to max setting no voltage change
should occur
5. Set inner range dial * 100
a. Press and hold'calibrate'
button
b. Adjust pot until data logger reads 10.0 volts
6. Re-check zero and balance
15
III. Calibrate Strain Channel
7. Instron strain data unit settings
a. Polarity = 1 (tension)
b. Range < 20
8. Attach extensometer to test sample
a. Place extensometer in closed position, but not restricted
b. Plug in extensometer cable into back of Instron load frame
9. Close channel 15 on data logger
10. Press and hold'zero'
button on Instron strain data unit
a. Adjust pot until data logger reads 0.0 volts
b. Release button
11. Adjust balance pot until data logger reads 0.0 volts
12. Set range switch to desired full scale range
a. Range = 5 for this work
13. Open extensometer to the equivalent full scale displacement
a. Be sure the extensometer is not outside of the test
portion of the sample at maximum displacement
b. 25% strain = 12.5 mm for this work
14. Adjust calibration pot on Instron strain data unit until
data logger reads 10.0 volts
15. Return extensometer to closed position and recheck zero
a. Improper seating of extensometer on sample will result
in voltages much greater than or less than 0.0.
16. Re-adjust balance to 0.0 volts as required
IV. Ready to Test
16
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21
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23
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4.0 EXPERIMENTAL RESULTS
4.1 Repeatability
Variation of test results was a prime concern for this study. As
previously stated, an alternative to machine crosshead displacement is
critical for more accurate test data.
The extensometer provided a very repeatable response despite the need to
re-attach it to each test sample. Figure 9 shows the transition region of
the polystyrene's tensile behavior for three runs at the same strain rate of
0.16%/second, which corresponds to a 5 mm/minute crosshead speed. Maximum
deviation is on the order of 1000 kPa at the yield point, which occurs at
approximately 1.25% strain and 28,000 kPa.
As the test speed increases, the yield point moves to a higher stress and
strain level but still with good correlation. Figures 10 and 11 exhibit
these trends for two different strain rates.
An overview of all extensometer test results is shown in Figure 12. As
expected, the maximum stress level rises with crosshead test speed. However,
the yield point is contained in region between 1% and 2% strain. Also, the
elastic behavior of the material does not vary significantly with speed, as
seen by the similar slopes of all curves in Figure 12.
25
When the strain gage data is examined, some differences from extensometer
results are immediately apparent.
As seen in Figure 13, the yield stress increases with the test speed, but
the yield strain remains nearly constant. This is where the similarity to
extensometer data ends. Note that in this figure the test speeds expressed
in % strain/second correspond to a 6.35 mm gage length.
Figure 14 illustrates both extensometer and strain gage data from test
runs at a 5 mm/minute crosshead speed. The yield stress still occurs at
approximately 28,000 kPa, but the gage yield strain is 3% versus 1.5% for the
extensometer yield strain. Also, the slopes are different, implying a
difference in modulus.
4.2 Difficulties
An extensometer is relatively simple to use. It is necessary to avoid
binding the knife edges on the sample, but this becomes easier with practice.
Also, to initially set the data logger before any acquisition begins, the
operator must physically open the extensometer to its maximum position, then
set the upper voltage limit on the data logger to 10 volts. This will match
the data logger full-scale output to the extensometer bridge full-scale
output. Voltage response at intermediate extensometer positions was found to
be linear. Also, it was not necessary to reset the full -scale voltage for
each test run.
26
Foil strain gages presented a number of minor irritants. Gage
application, per se, was not a problem. However, attaching lead wires to the
gage was difficult. Care must be taken not only to avoid damaging the gage
itself but also to keep the soldering iron away from the test sample.
Excessive heat will cause localized melting of the polymer and render the
sample useless.
In addition, strain gages offer a limited range for testing, typically 5%
strain or less. This may present a problem for some polymeric materials.
The output of the digital strain indicator is also restricted to low strain
ranges. Some variability in the output of the VE-20 strain indicator was
noted, and this contributed to the inconsistency of the strain gage data.
Generally, while strain gages may offer improved accuracy in some cases,
the extra preparation time can increase overall test time. Extensometers can
provide accurate results in a shorter time, thus allowing the option to test
more samples.
27
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(v0i * -ed^) SS3hLLS
Figure 10
Extensometer Tes: St.uin at 50 mm/minute
29
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Figure 11
xt i nsomeirer Test Results at . ,m:r> i n u L e
3 0
CO
NT
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cn
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10
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Figure 12
Extensometer Data from .;our Te-,c Speeds
31
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F i q a r e 13
Strain jaqe Test Data from 5 mm/min and 10 mm/mi:
32
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Fiqure 14
Strain Gage vs Ex: nso,T,r?ter Test Results at 5 mm/mm
33
5.0 VISCOELASTIC THEORY
Viscoelastic theory describes those solids which exhibit both elastic
deformation and viscous flow; a joint fluid-solid behavior. The
deformation process has a time dependent component that can have a
significant effect upon the material's response.
Two major forms of constitutive equations exist for viscoelastic
materials, integral and differential. The differential form can be
represented by a mechanical model composed of a series of springs and
dashpots. Linear viscoelasticity implies that the stress is proportional
to the strain at a given time and that superposition is possible. Also,
the assumptions of homogeneity, isotropy and small strains are included
(3,4,5,6).
The two most fundamental mathematical models are the Maxwell Fluid
and the Kelvin Solid, which serve as the basis for more complex systems.
5.1 Maxwell Model Characterization
The Maxwell Model has one linear spring and one viscous dashpot in
series, as illustrated in Figure 15:
rVWit
K
Maxwell Fluid
Figure 15
3 4
The total strain is total= =
Gspring + dashpot (5_1)
For this system, the spring and dashpot experience the same force or
stress. Thus, the element relations are:
<y = t_spring (5-2)
"
=
1^
dashpot (5"3)
at
where: cr = stress
. = strain
^ = k. = first derivative with respect to time
E =
spring constant
n =
viscosity constant
To obtain the system's equation of motion, perform the following
steps:
Differentiate (5-1):=
spring+ dashpot (5~4)
Differentiate (5-2): cV = E .spring (5-5)
Substitute (5-3) & (5-5) into (5-4):
+ (^K (5-6)
(5-7)
35
The deformation behavior can be obtained by evaluating this element
for two standard load environments. For the first case:
(a) Strain response due to constant stress, applied
instantaneously at time t - 0
Initial conditions for (a) are:
2J-t
Thus, equation (5-7) becomes:
(5-8)(|-Vo - Ei
1
Rewriting:
Integrating:
1
(5-9)
(5-10)
Where the integration constant can be represented by the strain at
time t = 0, written as C0 .
c /-A - <K + +
36
These relations indicate an increase of strain under constant stress,
a condition known as creep (5> 6) and displayed in Figure 16 for the
region between 0<t<t, .
f
-"L
Maxwell Creep Behavior
Figure 16
The second load case involves:
(b) Stress response due to a fixed strain at time
t = ti, ti 0
The initial conditions are now:
a c(-t,,
a-t
= o
Thus, for a fixed strain, the system equation (5-7) reduces to:
/ F ^
cy-+ }<y = o
(5-12)
37
Integrating (5-12) results in a solution of the form:
Where <yo is the stress at time t=0
(5-13)
Rewriting:
o-(-t> = cr-(VO
(5-14)
where:
T-t
aXc
Equation (5-14) illustrates a decrease of stress under constant
strain, known as stress relaxation (5>6) Figure 17 shows this behavior.
Maxwell Relaxation Behavior
Figure 17
It is important to note that this system exhibits a single relaxation
time, generally written as Tau (t) .
3j
5.2 Kelvin Model Characterization
A Kelvin or Voight solid has one linear spring and one viscous
dashpot in parallel, seen in Figure 18.
hvwv-
EH
Kelvin Sol id
Figure 18
For this system, the elongation or strain on both elements is the
same, thereby requiring that the total force or stress be written as
(5,6):
totalc? = cr
spring+
"
dashpot (5-15)
Substitute the element relations from (5-2) and (5-3) and drop the
subscripts on to obtain the system's equation of motion:
cr - Ze. -+ n (5-16)
39
The mechanical response of this system can be characterized by
applying the same conditions as for the Maxwell Fluid.
Due to a constant stress at time t = 0, equation (5-16) yields:
cr - E +
V (5-17)
rewriting:
^ 1
(5-18)
The solution to this ordinary differential equation has the form:
- (V) .
(-tV
| (l- C ) (5-19)
where:
y*
From equation (5-19), a gradual increase in strain emerges until a
finite value is reached, a condition known as delayed elasticity and
shown in Figure 19.
<Y.
-I
Kelvin Delayed Elasticity
Figure 19
140
For a constant strain applied at time t = t, the system response of
equation (5-16) becomes:
<x(~,)~ <r-
E, (5-20)
Using equation (5-19) with t = t, provides ,
. ^ (lE
-^-/r)
(5-21)
Substituting (5-21) into (5-20) gives:
(Vt)<*"--
; \-
e (5-22)
In this case, an initial decrease of stress occurs, but the stress
remains at this reduced level. This is known as incomplete relaxation
and is illustrated in Figure 20.
Kelvin Relaxation
Figure 20
Further analogies can be made by adding elements to the two basic
ones described above. Each can be useful depending upon the particular
material under investigation. Table 3 provides details on some of the
more common models built on these two concepts (3,5,6).
41
TABLE 3
VISCOELASTIC MODELS
MODEL TYPE DESCRIPTION SYSTEM EQUATION
Maxwell Fluid Spring & Dashpot in Series C +pcr =
a .
Kelvin Solid Spring & Dashpot in Parallel <r - + q .%.* +
1
Maxwell -Wei chart Maxwell Elements in Parallel -. x 2. (o- + p, b^)
I
Voight-Kelvin Kelvin Elements in Series cr - *(%, +%.^
Three Parameter Spring Plus Kelvin Solid in
Solid Series
Four Parameter Spring Plus Dashpot Plus
Fluid Kelvin Solid in Series
c +-
42
Chapter 6.0
Brinson Viscoelastic Model
6.1 Description
The viscoelastic elastic model proposed by H.F. Brinson (15) is also
known as the modified Bingham Model. For this model, the yield point is
defined as the point of maximum engineering stress. In addition, an
elastic limit is established as the transition point from linear elastic
behavior to linear viscoelastic behavior. On an engineeringstress-
engineering strain diagram, this limit, identified as, emerges as the
first point at which nonlinearity occurs.
For this model, two equations are required:
cr-Ze,
<r- &
(6.1}
which is merely Hooke's Law and:
^&+TER\l- / (6-2)
which defines the viscoelastic behavior beyond the transition point &.
43
Parameters from equation (6-2) are:
= elastic limit stress
dp = elastic limit strain
El = elastic modulus
R = strain rate
t = relaxation time
It is important to note that only a single relaxation time resides in the
above relationship.
As Brinson discussed, the model has some limitations in
characterizing the behavior of some polymers, especially those with
moduli that are strain rate dependent. However, the model can be
utilized to predict the stress-strain response due to a change in strain
rate.
6.2 Computer Adaptation of Model
Examination of equation (6-2) yields four variables for any known
strain rate: ,<>, E and t. One way to implement this model is to
obtain some actual constant strain rate test data. From these results,
assuming that no large sample-to-sample deviations exist, an estimate can
be made for the elastic modulus (E) and the transition point coordinates
(@,4>). With these parameters defined, the relaxation time (t) can be
modified in order to obtain the best fit of the data.
44
Calculation of the elastic modulus is straightforward for most
polymeric materials. The definition of the elastic limit transition is
subject to wide variation.
Ideally, all stress variations due to changes in strain rate would
appear to converge as the percent strain approaches zero. Brinson
presented some polycarbonate data of this vein, as shown in Figure 21.
Actual tensile results for the high impact polystyrene tested for this
thesis are seen in Figure 22 for machine crosshead speeds from 0.5
mm/minute to 10 mm/minute. No significant deviation from linear behavior
is evident until the yield point is reached, which is a typical response
for brittle polymers.
Expansion of the elastic-viscoelastic region does not uncover any
surprises (Figure 23). As the higher strain rates are investigated, some
minor differences start to emerge (Figure 24). Note that the strain
rates on this plot span one and one-half decades.
For the present analysis, experimental test results from three
machine speeds (1, 5, 500 mm/minute) provided an average elastic modulus
of 2.34 E6 kPa. The elastic to viscoelastic transition point was
difficult to determine explicity. A value of 0.9% strain was chosen for
^ to allow for viscoelastic effects in the model prior to reaching the
yield point. The corresponding stress for e is approximately 21000 kPa.
45
With the definition of these parameters, the value of the relaxation
time, T, governed the match between experimental data and theoretical
calculations. A least squares analysis was used to judge how well the
viscoelastic model fit the actual results. Two statistical quantities
provided a measure of comparison, normalized error (kPa per percent
strain), and overall root mean square error (kPa).
6.3 Comparison to Experimental Results
Data from two extreme crosshead speeds of 1 mm/minute and
500 mm/minute were selected to find the appropriate relaxation time in
seconds. Figure 25 shows the best correlation for the lower speed, with
an rms error of only 600 kPa, for a tau of 2.0 seconds. Table 4 presents
the results in tabular form.
Similarly, the higher crosshead speed data produced a value for tau
of 0.31 seconds. This correlation was more difficult because there are
only two data points that exist below the yield strain. Consequently,
the initial portion of the stress-strain curve suffers, as seen in Figure
26. Any attempt to match the initial slope more closely manifests itself
in gross errors for the post yield region. Thus, while Table 5 shows a
low normalized error, the rms error is much larger than for the
1 mm/minute speed.
46
In order to predict deformation behavior at other strain rates, a
relationship between tau and strain rate is necessary. The simplest
format is a linear relation, constructed from the values of tau described
above. Using the data from the two test speeds described above, the
result is:
T =-0.1184 + 2.00 (6-3)
The value of tau was then predicted using equation 6.3 for a strain
rate corresponding to a machine crosshead speed of 5 mm/minute, with the
result shown in Figure 27 and Table 6. While the viscoelastic model
predicts an accurate yield stress, it shows a nearly constant bias error
for the post yield behavior. This is in contrast to how the experimental
data was initially matched. As the strain rate increases, the overall
rms error will continue to increase, as evident in Figure 28. Table 7
exhibits a rms error 35% larger for a doubling of the strain rate.
Obviously, for higher strain rate data, the error will be unacceptable.
Due to the nature of the Instron Universal Test Machine, a
significant change in crosshead speed occurs near the maximum machine
capability. Specifically, 500 mm/minute is the highest speed, with
100 mm/minute as the next lower speed. Therefore, to preclude any
discrepancies that might exist, the experimental data from the
100 mm/minute test was matched directly, which resulted in a tau of 0.10
seconds (Figure 29). Another linear equation was calculated using this
new tau as the upper endpoint, giving:
47
T =-0.575 + 2.02 (6"4)
Again, the viscoelastic model output is examined against 5 mm/minute
test data (Figure 30). The shape is no different than the one shown in
Figure 26, while Table 8 indicates even larger errors.
6.4 Conclusions
A linear function is inadequate to predict the deformation behavior
at intermediate strain rates. However, for any given strain rate, the
magnitude of tau can be modified in order to obtain good correlation of
the model and experimental data.
This type of individual strain rate evaluation was done for four
additional strain rates, using similar criteria to that above for
determining the best match of model to data. Results are given in
Table 9.
In summary, the Brinson viscoelastic model can describe the tensile
deformation behavior over a wide range of test speeds, with limited
prediction capability.
48
1 l 1 1"
12 j3000/i. in/in/sec
^- 700/i in/in/sec
10
^80/Jt in /in/sec
^ #20//. in/in/sec
8
t/>
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co
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6ill
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41 r v PERIMENT/ C.A
/ M--B MODEL
2
1 1 1 1
80
60
o
Q_
CO
CO
I-
CO
- 20
0 2 4 6 8
STRAIN (%)
Stress-strain-strain-rate response
of PMMA
Figure 21
Brinson'
s: -
h K D s t a
49
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(3 (X)
ro ro
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(v0l * d>D SSBdiS
Figure 22
Extensometer Test Results for Slow Crossheaa Speeas
50
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['
iqure 2 3
Exten.ometer Data at Slow Speeds, iieid Point Region
51
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Figure 2&
Extensometer Data at Fast Speeds, Yi.J.d Point Region
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Final Fit of Brinson Model to 1 mm/min Data
53
BRINSON MODEL
RESULTS FOR STRAIN RATE OF
AND FOR AN ESTIMATED TAU OF
0333 X PER SECOND
19.96E-01 SECONDS
C RUN 0 j
STRAIN CX>
.12
.23
.33
.43
.53
.63
.73
.83
.93
1 .04
1 .15
1 .31
1 .51
1 .70
1 .89
2 .09
2 .29
2 .49
2 .69
3..72
4 .79
5..88
6..98
8.,09
9. 19
10. 30
11. 39
12. 48
13. 55
14. 63
15. 67
16. 70
17. 72
18. 73
EXPERIMENTAL STRESS CkPo}
3168.49
5766.40
8244. 74
10680.69
13057. 14
15385.11
17644.22
19829.54
21937.57
23903.90
25222.59
24733.05
23833.63
23278.51
22986.90
22818.26
22708.17
22630.88
22575.84
22444.67
22422.42
22443.50
22491.52
22554. 76
22629.71
22722.23
22815.92
22914.29
23025.55
23136.81
23238. 70
23353. 4 7
23458.87
23561.93
CALCULATED STRESS CkPaj
2781.79
5315.78
7643.03
10003.50
12447.63
14716.85
17088.44
19450. 08
21596.11
22356.31
22516.52
22552.00
22555.21
22555.37
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
22555.38
LEAST SQUARES ANALYSIS:
ERROR NORMALIZED TO STRAIN CkPa/X) ROOT MEAN SQUARE ERROR CkPa)
34.33E+00 62.50E+01
Table
54
(-01 * Bd>l) SSBdlS
F 1 a / 6
Final Fit of Brinson Mo '1 -o 500 mm/min Data
X*
2i i
CL
\~
LO
O !
Ul
<
ui
a
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x
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u
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m tt
n UJh-
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55
BRINSON MODEL
RESULTS FOR STRAIN RATE OF 16.6666 X PER SECOND
AND FOR AN ESTIMATED TAU OF 31.00E-03 SECONDS
C RUN 2 j
STRAIN CX> EXPERIMENTAL STRESS CkPaj CALCULATED STRESS CkPa)
.82 25331.50 19082.70
1.31 33646.51 27574.61
2.02 34958. 17 31696.36
2.70 34009.56 32719.12
3.18 33587 . 95 32944.13
3.53 33318.59 33015.89
3.89 33142.93 33052.51
4.25 32990.68 33071.48
4.62 32850.14 33080.93
5.00 32721.32 33085.60
5.38 32615.92 33087.90
5.77 32522.23 33088.97
6.16 32440.25 33089.49
6.55 32369.98 33089.74
6.95 32299.71 33089.85
7.33 32241.16 33089.90
7.73 32194.31 33089.93
8.12 32147.47 33089.94
8.52 32100.62 33089.95
8.92 32065.49 33089.95
9.31 32030.35 33089.95
9.70 32006.93 33089.95
10. 09 31971.80 33089.95
10.48 31960.09 33089.95
, 10.88 31936.66 33089.95
11.27 31924.95 33089.95
11.67 29793.50 33089.95
11.67 14041.83 33089.95
LEAST SQUARES ANALYSIS:
ERROR NORMALIZED TO STRAIN CkPa/X) KOOT MEAN SQUARE ERROR CkPa)
52.41E+00 39.53E+02
Table 5
56
(-01 * ed^1) S53dlS
2ii
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LD
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Hui
2
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to UI
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III ui
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in l-l
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Figure 27
Result of Predicting Tau Linearly at 5 mm/rain
57
BRItlbON MODEL
RESULTS fuK bTKHlN kAlt OF
AND FOR A hKtDlurtD I HU OF
1666 X PER SECOND
19.80E-01 SECONDS
C RUN 1 )
RAIN CXi EXPERIMENTAL STRESS CkPa; CALCULATED STRESS (kPa)
.15 3770.21 3482.21
.27 6594. 04 6299. 10
.38 9106.11 8795.48
.49 11679.08 11487.06
.60 14265.51 14003.15
.71 16737. 76 16582.41
.82 19149.12 19247.67
.93 21467.95 21668.36
1.03 23697.78 23539.4 0
1.15 25838.60 25055.52
1.26 2 7685.4 7 26121.08
1.43 28312.02 27186.48
1.69 2 7289.63 28020.63
1.97 26266.06 28418.34
2.24 25577.44 28588.04
2.50 25152.32 28660. 25
2.75 24875.93 28691.64
2.99 24703. 78 28706.46
3.24 24576. 12 28713.77
3.96 24354. 78 28719.52
4.67 24255.23 28720. 16
5.37 24227.13 28720.23
6.07 24220.10 28720.24
6.76 24231.81 28720.24
7.45 24251. 72 28720.24
8.13 24284.51 28720.24
8.81 24323. 16 28720.24
9.48 24362.98 28720.24
10.15 24415.68 28720.24
10.81 24462.52 28720.24
11.47 24516.40 28720.24
12.13 24578.47 28720.24
12.78 24633.51 28720.24
13.43 24687.38 28720.24
14.08 24/4 7. 11 28720.24
14.72 24802.15 28720.24
15.35 24868.91 28720.24
15.99 24926.29 28720.24
16.62 24987.19 28720.24
17.24 25050.43 28720.24
17.86 25113.67 28720.24
18.48 25182. 77 28720.24
19.10 25237.81 28720.24
19.71 25298. 71 28720.24
20.32 25361.95 28720.24
LEAST SQUARES ANALYSIS:
ERROR NORMALIZED TO STRAIN CkPa/X) ROOT MEAN SQUARE ERROR CkPa)
57.86E+0037.63E+02
Table 6
58
CD
Q
13
2
Ld
o i a
Ldr-
2
Ld
2>-
\-
ffi
>
_d
O
a.
OJ CDNi'
H LD
CO OJ OJ OJ '
CE -01 * Ed>l) SSBdlE
X
Ii
CT
12
10
z|-'
i i
-x
ui
ft
Ul
IX
X
UI
a
z
o
u
u
tn
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M UJ
nh-
UJN- i;
o
in
()
n u
c H
m X
UJ
a tr
I.I Ul
1,1 u
n _J
in u
Ul
H z
(n IL
hi ft
HI-
F i g u r..- 2 ::-
Brinson Model Predicted Re vonse at 10 mm/mm
59
BRINSON MODEL
RESULTS fuk SlkHiN KATE OF
AND FOR A KktUlortO TAU OF
3333 X PER SECOND
19.61E-01 SECONDS
C kUN 0 )
STRAIN CX) EXPERIMENTAL STRESS CkPa) CALCULATED STRESS CkPa)
.16 4833. 13 3854.51
.27 7420.38 6431.90
.37 9659.46 8733.47
.47 12079.02 11009. 12
.57 14603.97 13429.26
.68 16971.99 16003.84
.78 19286. 14 18350.28
.88 21505.43 20600.19
.98 23609.94 22833.71
1.08 25633.65 24706. 10
1.19 27440. 70 26436.71
1.32 28500.57 28246.56
1.51 27953.66 30295.61
1.73 26988 . 65 32008.73
1.94 26192.28 33182.85
2.15 25612.57 34048.27
2.35 25224.93 34636.02
2.56 24954.40 35078.40
2.75 24781. 07 35394.96
3.36 24443. 79 35937.08
3.98 24288.03 36153.49
4.59 24207.22 36237.96
5.23 24180.28 36271.29
5.86 24168.57 36284.0 0
6.50 24188.48 36288.82
7.15 24210. 73 36290.62
7.79 24247. 04 36291.30
8.43 24288.03 36291.55
9.06 24321.99 36291.64
9.69 24370. 00 36291.68
10.32 24419. 19 36291.69
10.94 24464.87 36291.70
11.51 24515.22 36291.70
12.09 24566. 75 36291.70
12.71 24626. 48 36291.70
13.31 24681.52 36291.70
13.90 24740.08 36291.70
14.49 24798.64 36291.70
15.06 24859.54 36291.70
15.65 24916.92 36291.70
16.26 24964.94 36291.70
16.86 25018.81 36291.70
17.46 25078.54 36291.70
18.04 25133.58 36291. 70
18.63 25197.99 36291.70
LEAST SQUARES ANALYSIS:
ERROR NORMALIZED TO STRAIN CkPa/Xj ROOT MEAN SQUARE ERROR CkPaD
16.94E+01 10.72E+03
Table /
60
TT"
tn
Q
2o
CJ
ui
in
2OJ
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i vr
ro
LjJta
_j
L.
M o
CO n
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h~ ui1-
IX
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m
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n
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C.i; 2-v O^
cn
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ft
2ro
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r-
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CD
CO
OJ CD ^r O CD OJ CD
ro ru oj oj
(-0 1 * Bd>1) SS3diS
\r 2
z'-' ii
Ul
X
l-H
22
2 O
r-
o
ui
t/Jtn
ft ftUl ua.
k 2O
n 2cn Uln i
m XUl
n
..
ftn U!Ul uUl :jft atn cn
2i-
tztn ftUl 1-
H
Fie are -9
Final Fit of Brinson Model to 100 mm/min Data
61
tn
2!2
O
"2 Ul
2(J!
2 GJ
Ul
i<i-
CVJ
LdCTJ
ll
n O
cn :j
2!
,1
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Ld Q.
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2JC
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Zi-i
<D ck
2ro
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a:
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cn
CJ
utn
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Q.U
iso
tn
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or
2UJ
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i.: o
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in Ul
CI)t- z
tn ft
UJ ft
(-01 x ^d>!) SS3dlS
Figure 30
Result of Second Linear Prediction at 5 mm/mm
62
BRINSON MODEL
RESULTS FOR STRAIN RATE OF .1666
AND FOR A PREDICTED TAU OF 19.24E-01
PER SECOND
SECONDS
C RUN 0 >
STRAIN CX5 EXPERIMENTAL STRESS CkPai CALCULATED STRESS CkPaJ
.18 4193.11 4278.98
.29 6670.86 6713.46
.39 9194.41 9097.34
.49 11680.01 11516.31
.60 14150.74 13926.51
.70 16516.42 16436.16
.80 18831.74 18707.72
.90 21056.88 21059.42
1.01 23214.10 23120.41
1.11 25235.47 24578.33
1.21 26978.11 25684.19
1.36 27505. 11 26731.99
1.55 26458.13 27499.03
1.71 25446.27 27898.98
1.85 24840.80 28118.50
2.00 24476.58 28261.53
2.15 24241.18 28350.21
2.31 24073.71 28408.25
2.47 23961.28 28444.56
3.30 23693.09 28497.13
4. 19 23624.00 28501. 14
5. 13 23624.00 285 01.38
6.11 23659.13 28501.40
7.10 23710.66 28501.40
8.11 23778.59 28501.40
9. 12 23855.88 28501.40
10.14 23946.06 28501.40
11.17 24030.380
28501.40
12.20 24127.58 28501.40
13.23 24217.76 28501.40
14.25 24316.13 28501.40
15.27 24416.85 28501.40
16.28 24519.91 28501.40
17.29 24612.43 28501.40
LEAST SQUARES ANALYSIS:
ERROR NORMALIZED TO STRAIN CkPa/X) ROOT MEAN SQUARE ERROR CkPa)
78.97E+00 41.45E+02
Table 0
63
TABLE 9
Brinson Model
Results of individual fit of data from different strain
rates
(No Prediction)
Strain Rate (%/Sec.) Tau (Sec.)
0.03 2.00
0.16 0.950
0.33 0.555
1.66 0.100
3.33 0.088
16.66 0.031
64
CHAPTER 7.0
MATSOUKA VISCOELASTIC MODEL
7.1 Description
S. Matsouka (16) developed a viscoelastic model which evolved from a
basic Maxwell Model having a single relaxation time. For the Maxwell
model, the relaxation modulus takes the form:
(7-1)
-(-LAO
where: E0 = initial elastic modulus
t0= relaxation time constant
The corresponding stress distribution can be written as:
x
For polymers, the simple exponential relationship for E(t) produces a
result which is much broader than seen from test data. To account for
the sharpness in the relaxation behavior, equation (7-1) was modified to
obtain the William Watts equation:
65
where:
= empirical constant with value less than 1
"i", r X relaxation constant
The effect of adding the P> term is illustrated in Figure 31.
Matsouka found that relaxation in polymeric systems exhibits non
linear viscoleastic effects that depend upon the strain rate. Thus, an
additional term was added to the William Watts equation to account for
the strain effect:
E(t.O = E/O e. (7-4)
= E&e c (7-5)
where: = strain
0 = strain at the yield point
The total stress is given by the convolution integral:
R, ^ f EC-t-X O if dX(7"6)
Substituting from equation (7-5) for E (-t,c) into (7-6) gives:
dX
66
where: X = time
-t= variable for the integration
Note that the shift due to convolution has no effect on the purely
strain dependent term.
It is desired to transform the stress equation from a function of
time to a function of strain. To begin, multiply the second exponential
term of (7-7) by to obtain:
*-(-t)= \ Ee e e. d(7-8)
6
For a constant strain rate, the strain can be described as:
strain = strain rate x time
or
. = IX (7-9)
Matsouka's final result can now be written:
-() = \ Ee C tUrJ (7-10)
o
where:= C"t
C = strain rate
= strain at the yield point
P = empirical constant
E."
T = adjustable parameter
67
7.2 Computer Adaptation of Model
Examination of equation (7-10) reveals four parameters needed to
implement this viscoelastic model for a given strain rate:
P ,E0, e , and Ti .
As Matsouka explains, P> is a temperature dependent empirical constant
which is always less than unity. For room temperature, an approximate
value for (3 is 0.03, with more detailed calculations for polystyrene
resulting in a value of 0.0242.
The parameter E0 is no longer simply the initial elastic modulus, but
is related to the slope of the relaxation modulus. However, since the
relaxation behavior depends upon the applied strain rate, E0 is also
strain rate dependent.
A direct calculation of this parametric value would involve numerous
stress relaxation tests at different strain levels. From this matrix of
data points, the value for E0 could be extrapolated. Stress relaxation
tests are difficult to run, especially on a test machine without a
feedback loop to maintain a constant load. Some tests were attempted,
with mixed success.
Rather than develop another entire test plan, the value for E0 of
585 ksi was extrapolated fromMatsouka'
s data for polystyrene.
68
The third parameter, 0 , is the strain value at the maximum or yield
stress of the polymer. Its value needs to be determined experimentally
from constant strain rate tensile tests. From the test data illustrated
in Figures 16 through 18, a value of e= 1.5% was chosen. With the above
quantities specified, the magnitude of *T| determines the correlation of
theoretical results to experimental data.
To perform the integration numerically, Simpson's rule was employed,
in the form:
x 1f( + (2xF6)+ (4yFt)+ F,l
I 3 (7-11
1= Af
(7-11)
where:
AC =
step size
Fj = value of integrand at the first data point
F2 =" "
last data point
F0 = sum of the integrand values at all odd numbered
steps
Pc =" "
even numbered
steps
In this analysis, the step size was made dependent upon the magnitude
of the strain. For all strain values equal to or less than the yield
strain, ,the area between the first data point and the current data
point was divided into ten sections. This resulted in a step size of:
A- Vo (7"'2)
69
Since the extensometer provided useful post-yield strain data, the
number of sections was increased to twenty to preserve the accuracy of
the calculations.
The same least squares relationships as described in Section 3 were
utilized to assess the correlation of the model to the test data.
7.3 Comparison to Experimental Results
As was done for the Brinson model, experimental data from a cross-
head speed of lmm/minute was selected for the initial correlation.
However, due to the nature of the polystyrene, a problem immediately
emerges, as seen in Figure 32. The initial slope of the curve cannot be
matched without seriously overshooting the constant strain region above
4% strain. Consequently, the smallest rms error occurs when the visco
elastic model matches the post-yield behavior. Table 10 provides the
corresponding numerical values, with an overall rms error of 2974 kPa,
for a tau, of 2500.
70
For the higher crosshead speeds, this effect is magnified as shown in
Figure 33 for the lOOmm/minute rate. The best correlation in this case
has an rms error of 5031 kPa, with a tremendous increase in the value of
tau,, now equal to 500,000 (see Table 11).
The objective was to again develop a relationship between the test
speed and the adjustable parameter tau,. However, with a basic
difference in the shapes of the experimental and viscoelastic
representations, an accurate prediction for other test speeds seems
unlikely.
For consistency, a linear relationship for tau was assumed, using the
values found for the lmm/minute and lOOmm/minute speeds:
<Tt= ( i.si E.+S) .
- 2520.
(7-13)
A prediction was made for the lOmm/minute test speed, with the
results shown in Figure 34. As expected, the desired correlation of the
model to the test data was not achieved, as given by the 3528 kPa rms
error of Table 12.
According to Dr. H. Ghoneim (personal communications and unpublished
manuscripts), an additional term would improve the viscoelastic model.
Specifically, an exponent, . > 1, added to the strain dependent term of
equation 7-10, would provide a response with a sharper post-yield
decrease in stress level. The governing equation now becomes:
o-<0=
) E* e e. <4 (7-14)
71
The effect on the viscoelastic model response is quite significant,
as illustrated in Figure 35 for the lOOmm/minute data. As well as a
steeper initial slope, some post-yield decrease is evident for the
relatively small value of |2>0= 3. Table 13 shows dramatic improvements
in both correlation measures, including a 53% decrease in the rms error.
Larger values for p result in a model with sharper post-yield response,
which can lead to instability or oscillatory calculations at higher
strain levels.
7.4 Conclusions
The Matsouka viscoelastic model does not accurately represent the
behavior of brittle polymers with large post-yield decreases in tensile
strength. However, the model is quite powerful and would be applicable
to many other polymers.
The addition of the . term will improve the correlation of the
Matsouka model to the experimental data for polymers similar to the high
impact polystyrene of this study. Some consideration should be given to
computer algorithm stability.
72
< 12- >
I
I
Is
o
A
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/
a;./...:...
>
rt- ~~r~
Ci
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i
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t
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y
s
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- r}-
r-
o o
111
o
o
CO
o
O(N
o
UJ
Zi-
CD3 'aamndwv
Fiai'.Uf 31
Comparison of Maxwell and Wixiiam-Watts Viscoelastic Models
73
s
LY
U
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i i
cn
z
u
o
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cm
+
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in
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Figure 32
Final Fit of Matsouka Model to 1 mm/min Data
74
MAIbOUKA MODEL
RESULTS FOR STRAIN RATE OF .0333 Z PER SECOND
AND FOR AN ESTIMATED TAU OF 25.00E+02 SECONDS
C RUN 0 )
RAIN CIO EXPERIMENTAL STRESS CkPa) CALCULATED STRESS
. 12 3168.49 2110.56
.23 5766.40 3838.68
.33 8244.74 5299.13
.43 10680.69 667 0.18
.53 13057. 14 7983.44
.63 15385.11 9113.83
.73 17644.22 10210.97
.83 19829.54 11224.39
.93 21937.57 12164.79
1.04 23903.90 13067.50
1.15 25222.59 13934.42
1.31 24733.05 15107.44
1.51 23833.63 16164.57
1.70 23278.51 17225.22
1.89 22986.90 18137.84
2.09 22818.26 18951.57
2.29 22708.17 19666.41
2.49 22630.88 20278.78
2.69 22575.84 20821.17
3.30 22476.29 22038.15
3.94 22430.62 22825.96
4.57 22417.73 23306.43
5.23 22424. 76 23597.04
5.88 22443.50 23758.38
6.54 22473.95 23840.56
7.20 22502.06 23872.93
7.87 22541.87 23875. 11
8.53 22584. 03 23859.59
9. 19 22b29. 71 23834.50
9.86 22688.26 23804.28
10.52 22743.31 23771.98
11.17 22791.32 23739.28
1 1 . 83 22853.39 23707.39
12.46 22914.29 23676.35
13.12 22982.22 23646.99
13.77 23043. 1223618.90
14.41 23116.9023592. 13
15. 04 23178.9723567.39
15.67 23238.70 23544. 08
16.29 23312.4823521.86
16.91 23371. 0323501. 14
17.52 23436.6223481.62
18. 12 23502.2023463.21
18.73 23561.9323445.86
19.32 23629.8523429.58
(IcPa)
LEAST SQUARES ANALVSIS:
ERROR NORMALIZED TO STRAIN CkPa/X^ ROOT MEAN SQUARE ERROR CkPa)
33.84E+0129. 74E+02
ri a b J.
7 5
a
u
eg
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Firidx Fit c
Figure 33
Matsouka Model to 100 mm/mm Data
76
MATSOUKA MODEL
RESULTS FOR STRAIN RATE OF 3.3333 Z PER SECOND
AND FOR AN ESTIMATED TAU OF 50.00E+04
i RUN 1 )
RAIN CZ) EXPERIMENTAL STRESS CkPa) CALCULATED STRESS
.28 10102.15 5483.92
.55 16407.50 9744.15
.83 22661.33 13354. 15
1.09 28458. 41 16197.47
1.38 32569. 07 18661.42
1.79 32112.33 21324.97
2.31 31093.45 23850.20
2.81 30402.49 25546.39
3.29 29922.32 26681.78
3.77 295 70.98 27497.91
4.25 29336. 76 28064. 08
4.74 29137.67 28459.04
5.23 28997. 13 28727.64
5.72 28891. 73 28906.86
6.20 28786.33 29022.56
6.69 28727. 77 29094.89
7.18 28692.64 29135.76
7.67 28645.79 29155. 11
8.15 28622.37 29159.65
8.64 28598.95 29154. 17
9.12 28575.53 29142. 11
9.61 28575.53 29125. 47
10.10 28575.53 29106.28
10.58 28563.81 29085.52
11.06 28587.24 29064.21
11.55 28587.24 29042. 43
12.03 28587.24 29020.66
12.51 28610.66 28999.58
CkPa)
LEAST SQUARES ANALYSIS:
ERROR NORMALIZED TO STRAIN CkPa/Z) ROOT MEAN SQUARE ERROR CkPa)
28.72E+01 50.31E+02
Table 11
77
(S
z
n
a+
zui
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f^
1T
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z t '
uh-
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Result of a Linear Prediction for Tau at 10 mm/mi,
70
MATSOUtCA MODEL
RESULTS FOR STRAIN RATE OF .3333 Z PER SECOND
AND FOR AN PREDICTED TAU OF 47.81E+03
C RUN 0 }
STRAIN CZ) EXPERIMENTAL STRESS (kPa) CALCULATED STRESS CkPa)
.16
4833.13 3142.54
.27
7420.38 5008.26
.37
9659.46 6545. 75
.47
12079.027958.09
.57
14603.979352.19
.68
16971.9910723.03
.78
19286. 1411879.56
.88
21505. 4312911.62
.98
23609.9413940.95
1.08 25633.6514849. 07
1.19 27440. 7015759.73
1.32 28500.5716807.13
1.51 27953.6617962.53
1.73 26988.6519293.26
1.94 26192.2820379.74
2.15 25612.5721330.38
2.35 25224.9322093.54
2.56 24954. 4022769.01
2.75 24781. 0723337.05
3.36 24443. 7924645.51
3.98 24288. 0325489.60
4.59 24207. 2226017.46
5.2324180.28
26345. 71
5.8624168.57
26537.35
6.5024188.48
26641.50
7. 1524210. 73
26689.99
7.7924247. 04
26704.31
8. 4324288. 03
26697.94
9. 0624321.99
26679.79
9.69
10.32
10 . 94
24370. 0026655.22
24419. 1926626.94
24464.8726597.77
11.51
12.09
12.71
24515. 2226570.78
24566.7526543. 74
24626.4826515.67
26489.37
13.31
13.90
24681.52
24740. 0826464.50
26440.86
14.49
15.06
24798.64
24859.5426419. 03
26397.66
15.6524916.92
26376.33
16.2624964.94
26356. 72
16.8625018.81
26337.97
17.4625078.54
26320.58
18.0425133.58
26304. 01
18.6325197.99
LEAST SQUARES ANALYSIS:
,^cn to qtbain CkPa/X) ROOT MEAN SQUARE ERROR CkPa)
ERROR NORMALIZED TO STRAIN CkPa/iJ
28E+02
33.69E+01
Ze
7 9
z
z
I
Ld
zII
cn
z
Ld
H
U
z
u
z
zr~
if]
z
z
o
z
S) CD
CD
OJ
cn
CD
OJ OJ
SJ
OJ
CD OJ CO S3
X
_J
J XX HO 2n Uld- z:UJ 1-1
o u
u xX X
H UI
+ 1
+ I
CE
(Y2O
r~
U
UJ
IZtn
X X
Ul Ul
0.
Js*
o
tn
n z
ro Ul
my~
n X
ui
o
X
Q Ul
UJ u
UI _J
X u
tn U)
2i- X
(D X
UJ1-
(Ev0I * ^d>1) 5S3dlS
Figure 35
Matsouka Model Response at 100 mm/min with _iie po Exponent
80
MATbOUKA MODEL
WITH EXTRA BETA ZERO TERM
RESULTS FOR STRAIN RATE OF 3.3333 Z PER SECOND
AND FOR AN ESTIMATED TAU OF 75.00E+06
( RUN 1 )
A1N CX) EXPERIMENTAL STRESS CkPa> CALCULATED STRESS CkPa)
.28
10102. 15 6469.89
.55
16407.50 12433.68
.83
22661.33 18064.43
1.09 28458.41 22536.73
1.38 32569.07 25923.09
1.79 32112.33 28332. 16
2.31 31093.45 29032. 08
2.81 30402.49 28975.26
3.29 29922.32 28878.17
3.77 2957 0.98 28796. 18
4.25 29336.7628727. 70
4.74 29137.6728668. 01
5.23 28997. 1328615.56
5.72 28891. 7328568.79
6.20 28786.3328526.85
6.69 28727.7728488.35
7.18 28692.6428453.28
7.67 28645. 7928421. 08
8. 15 28622.3728391. 78
8.64 28598.9528364.91
9. 12 28575.5328340. 15
9.61 28575.5328315.42
10.10 28575.5328289.92
10.58 28563.8128262.80
11. 0628587.24
28235.22
11.5528587.24
28208.97
12. 0328587.24
28187.65
12.5128610.66
28175.27
LEAST SUUARES ANALYSIS:
ERROR NORMALIZED TO STRAIN CkPa/Z> ROOT MEAN SQUARE ERROR CkPa>
17.61E+0123.43E-02
Table 13
8.0 CONCLUSIONS
With respect to data acquisition, the following conclusions were
reached:
(a) Extensometer transducers provide accurate and repeatable
strain data for uni-axial tension tests of polymeric
substances
(b) Extensometers have advantages over foil strain gages in
terms of test sample preparation, maximum strain range
available and repeatability of results.
A second objective of this study was the identification and
investigation of potentially useful viscoelastic models. In this area,
the following conclusions may be drawn:
(c) The Brinson model is easily implemented and can provide
a good correlation to experimental tensile data.
(d) By using a nonlinear relationship for the parameter t,
the prediction capability of the Brinson model could be
enhanced.
(e) Matsouka's convolution integral viscoelastic model produces
a smooth response that does not correspond well to experimental
tensile data for brittle polymers.
(f) The additional exponent ft, ,as recommended by Ghoneim (personal
communications), significantlyalters the Matsouka model
response and merits further investigation.
82
References
1. American Society for Testing of Materials (ASTM) Procedure D638M,
Volume 35, pg 259, 1982
2. ASTM Procedure D638 M, Volume 35, p. 250, 1982
3. C. G'Sell, J. Jonas, "Determination of the Plastic Behavior of Solid
Polymers at Constant True Strain Rate", Journal of Materials Science,
Volume 14, p. 583 (March 1979)
4. I. Gilmour, A. Trainor, R. Haward, "Elastic Moduli of Glassy Polymers
at Low Strains", Journal of Applied Polymer Science, Volume 23,
p. 3129 (1979)
5. M. Yokouchi, H. Uchiyama, Y. Kobayashi, "Effect of Tensile Strain
Rate on the Mechanical Properties of Polystyrene and High Impact
Polystyrene", Journal of Applied Polymer Science, Volume 25, p. 1007
(June 1980)
6. S. Matsouka, H. Bair, "The Temperature Drop in Glassy Polymers During
Deformation", Journal of Applied Physics, Volume 48, No. 10, p. 4058
(October 1977)
7. S. Sjoerdsma, D. Heikens, "The Constant Stress Behavior of Poly
styrene-Low Density Polyethylene Blends", Journal of Materials
Science, Volume 17, p. 2343 (1983)
8. S. Matsouka, H. Bair, S. Bearder, H. Kern, J. Ryan, "Analysis of
Nonlinear Stress Relaxation in Polymeric Glasses", Polymer Engineering
and Science, Volume 18, No. 14, p. 1073 (November 1978)
9. P. Naghdi, S. Murch, "On the Mechanical Behavior of Viscoelastic/
Plastic Solids", Journal of Applied Mechanics, Volume 30, p. 321
(September 1953)
References (Continued)
10. R. Schapery, "On the Characterization of Nonlinear Viscoelastic
Materials", Polymer Engineering and Science, Volume 9, No. 4,
p. 295 (July 1969)
11. S. Bodner, Y. Partom, "A Large Deformation Elastic-Viscoplastic
Analysis of a Thick Walled Spherical Shell", Journal of Applied
Mechanics, p. 751 (September 1972)
12. M. Rubin, "A Thermoelastic-Viscoplastic Model with Rate Dependent
Yield Strength", Journal of Applied Mechanics, Volume 49, p. 305
(June 1982)
13. S. Sjoerdsma, D. Heikens, "Model for Stress-Strain Behavior of
Toughened Polystyrene", Journal of Material Science, Volume 17,
p. 741 (March 1982)
14. V. Lubarda, "Correct Definition of Elastic and Plastic Deformation
and Computational Significance", Journal of Applied Mechanics,
Volume 48, p. 35 (March 1981)
15. H. Brinson, A. Dasgupta, "The Strain Rate Behavior of Ductile
Polymers", Experimental Mechanics, Volume 13, p. 458 (December 1975)
16. S. Matsouka, "Parametric Formulation of Viscoelasticity-Engineering
Properties of Molded or Extruded Plastics", Internal AT&T Bell Labs
Memorandum (1982)
17. W. Flugge, Viscoelasticity, (Blaisdell Publishing, 1967)
18. J. Ferry, Viscoelastic Properties of Polymers, (3rd Edition,
John Wiley, 1980)
19. J. Aklonis, W. MacKnight, M. Shen, Introduction to Polymer Visco
elasticity, (John Wiley, 1972)
20. W. Brown, Testing of Polymers, Volume 4 (John Wiley, 1967)
84
APPENDIX A
DATA ACQUISITION PROGRAM
FOR
DATA LOGGER
AND
INSTRON TEST MACHINE
85
10 ! ** PROGRAM FOR HP 3497A DATA LOGGER
20 ! ** CONFIGURATION: DVM 4 ANALOG INPUT ONLYAppendix A
21 ! WRITTEN BY L.T. LOGGI1 of 5
22 ! SEPT. 1984
23 !
TO OPTION BASE 1
< 0 CLEAR 7
50 PRINTER IS 1
60 PRINT USING",K"
70 PRINT"
DATA LOGGER SET-UP/DATACOLLECTION"
80 PRINT*
FOR THE INSTRON TENSILETESTER"
90 PRINT
100 PRINT "IT IS ASSUMED THAT TWO VOLTAGES AREPRESENT:"
110 PRINT*
STRAIN -
CHANNEL15"
120 PRINT"
LOAD -
CHANNEL16*
130 PRINT
140 PRINT "USE THE 2500 KG LOADCELL*
150 PRINT
160 PRINT "FOR TEST SPEEDS OF 20 MM/M1N ORGREATER,"
170 PRINT "YOU RUST HAVE A BNC JUMPER ONTHE"
180 PRINT"
REAR PANEL OF THE DATALOGGER,"
190 PRINT "BETWEEN THE 'VMCOMPLETE'
AND 'EXTERNALINCREMENT' PORTS"
200 PRINT
210 PRINT "PRESS'CONTINUE'
WHENREADY"
220 PAUSE
230 INTEGER Dial,Had
240 DIM Stroin(15C> ,LoadvC150) ,HsdataC60)
250 Factorl=980.0 ! NEWTONS/ VOLT
260 ! CAL LOAD ON 100 RANGE
270 Factor2-=2.5 ! MM/VOLT CEXT)
280 Factor3-.3175 ! MM/VOLT (GAGE)
290 Factor4-1.0 ! MM/VOLT CCHJ
300 Area-4.184E-5 ! M*2
310 Hsd-0 ! FLAG FOR HI TEST SPEED
320 Laet-0
330 Sf-0
340 Dial=0
350 Speed=0
360Cho*-""
370 ! INITIALIZE DATA LOGGER
380 ! DEFAULT DVM SETTINGS
390 OUTPUT 709$"SI"
400 OUTPUT 709;"TEO"
410 OUTPUT 709?"VF1*
420 PRINT USING"*,K"
430 PRINT "WHAT IS THE FULL SCALE LOADSETTING"
440 INPUT Lset
450 IF Lset =250 THEN Factorl =Fac tori/ - 40
460 IF Lset=100 THEN Factor l=Factorl
470 IF Lset=50 THEN Factor l=Factorl/2
480 IF Lset-20 THEN Factorl=Factorl/5
490 IF Lset-10 THEN Factor l=Factorl/10
5G0 IF Lset=5 THEN Factor l-Factorl/20
510 IF Lset<5 THEN
520 BEEP
530 PRINT
540 PRINT "FILTER ISREQUIRED"
550 PRINT "EFFECT ON THIS SOFTWAREUNKNOWN*
560 PRINT "SELECT ANOTHERRANGE"
570 WAIT 2.5
-j(. GOTO 420
390 END IF
,600 !/////////
610 PRINT USING "#,K"
620 PRINT "WHICH TYPE STRAIN MEASUREMENT?" Appendix A
630 PRINT 2 Of 5640 PRINT CD EXTENSOMETER"
650 PRINT "
C2> STRAIN GAGE"
660 PRINT"
'.3> BASED ON CH SPEED"
670 INPUT Sf
680 IF SfOINTCSf) OR Sf>3 THEN GOTO 420
690 IF Sf-1 THEN
700 PRINT USING ",K"
710 PRINT -WHAT IS THE EXTENSOMETER RANGE"
720 PRINT "
CDIAL SETTING ON INSTRON STRAIN DATAUNIT)"
730 INPUT Dial
740 IF Dial-10 THEN SfFactor2
750 IF Dial-5 THEN Sf-Factor2/2
760 IF Dial-2 THEN Sf-Factor2/5
770 IF Diol-1 THEN Sf -Factor2/10
780 IF DiaKl THEN Sf -Fac tor2/20
790 IF Dial-20 THEN
800 BEEP
810 PRINT
820 PRINT "INVALID SETTING-CHECK MANUAL"
830 WAIT 2.5
840 GOTO 700
850 END IF
860 Gl-50
870 GOTO 970
880 END IF
890 IF Sf-2 THEN
900 Sf-Factor3
910 Gl-6.35
920 ELSE
930 Sf-Factor4
940 Gl-127 ! ASSUME 5 INCH
950 END IF
960 ! ///////////
970 PRINT USING"#,K"
980 PRINT "WHAT IS THE TEST SPEEDCMM/MIN)"
S9C IKf'UT fptfcd
1000 CALCULATE S.'ACINC OF DATA FTS
1010 ! ASSUME A CONSTANT * OF PTS
1020 ! REGARDLESS OF TEST SPEED
1030 N-100
1040 IF Speed-. 2 THEN Tfactor=25.0
1050 IF Speed-. 5 THEN Tfactor-15.0
1060 IF Speed- 1 THEN Tfactor-10.0
1070 IF Speed-2 THEN Tfactor-5.0
1080 IF Speed=5 THEN Tfactor-2.0
1090 IF Speed-10 THEN Tfactor=1.0
1100 IF Speed>-20 THEN GOTO Ha_acq ! TO GOSUB
1110 Speed-Speed*Cl/60)*Cl/Gl)*100 ! CONVERT TO Z/SEC
1120 Speed- < INT(Speed*10000>>/ 10000
1130 !
1140 ! /////////////
1150 !
1160 PRINT USING"#,K"
1170 PRINT"
READY FOR DATA ACQUISITIONM*
1180 PRINT
1190 PRINT "START INSTRON AND COMPUTER AS SIMULTANEOUSLY AS POSSIBLE FOR ACC
URATERESULTS"
il0 PRINT
1210 PRINT 'PRESS'CONT:...^'
WHchREADY'
1220 PAUSE o-,
LOADCNEWTONS)"
1230 PRINT USING ",K"
1240 PRINT "WORKING*
*250 ! TAKE DATA DIRECTLY INTO ARRAYS
1260 PRINTER IS 701
1270 PRINT"
I DISPLACEMENT CMM)
l;::80 Start-TIMEDATE
i.'.90 FOR 1-1 TO N
1300 Etime-TIMEDATE-Start
1310 IF Etie<CTfactor*I) THEN GOTO 1300
1320 DISP "DATA PT #",1
1330 OUTPUT 709j"TE2AI15*
1340 ENTER 709, St roi n( I )
1350 Strain(I)-StrainCl)KSf
1360 OUTPUT 709j"AI16"
1370 ENTER 709;LoadvCI)
1380 LoBdv(i)-LoBdv(l)Factorl
1390 IF I>10 AND I <90 THEN GOTO 1440
1400 IMAGE 3D,8X,8D.4D,13X,8D.4D
1410 PRINT USING 1400 , I ,Strain( 1 ) ,LoadvCl )
1320 OUTPUT 709J'TE1IE"
1430 ENTER 709 j A
1440 NEXT I
1450 GOTO 1490
1460 ! /////////
1470 Ha_acq: GOSUB High_apeed
1480 ! /////////
1490 PRINTER IS 1
1500 PRINT USING"*,K"
1510 PRINT "DATA COLLECTIONFINISHED"
1520 WAIT .5
1530 PRINT USING"5/"
1540 LINPUT "DO YOU WANT TO SAVE THIS DATA ON DISK (Y/N)*,Cho*
1550 IFCho$-"N"
THEN GOTO 2050
1560 ON ERROR GOTO 1860
1570 PRINT USING",K"
1580 CAT
1590 PRINT
1600 PRINT "WHAT IS THE FILENAME FORSTORAGE*
161 0 iWPUT File
1620 IF Had-1 THEN
1630 CREATE BDAT File,30,30
1640 N-30
1650 ELSE
1660 CREATE BDAT Fi let , 100 , 100
1670 N-100
1680 END IF
1690 PRINT "PLEASEWAIT"
1700 ASSIGN #Pothl TO File*
1710 OUTPUT *Pathl,l;N, Speed
1720 OUTPUT *Pathl,2;Gl,Tfactor
1730 PRINT "N- ";N;" SPEED--;Speed;"
GL=*;G1
1740 FOR J-3 TQ N
i/50 ! CONVERT DATA TO X STRAIN
1760 ! AND KILO-PASCALS
1770 StrainCJ-2)-CCStrain(J-2))/Gl)*100
1780 LoadvCJ-2) -CLoadvCJ-2) /Area)/ 1000
1790 IF J<10 OR J>90 THEN PRINT USING 1400 ; J ,St ra i n<J-2) ,LoadvC J-2)
1800 OUTPUT #Pathl,J-,Strain(J-2),LoadvCJ-2)
1810 NEXT J
182" PRINT
,
R\^
"DATA ST" ^ED H FILE -->"jFiles.
\Z >5 BEEP 200,.2
1850 WAIT 2.0 88
Appendix A
3 of 5
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
2020
2021
2030
2040
2050
2060
2070
2080
2090
2100
2110
Z1ZQ
2130
2140
2150
2160
2170
2180
2190
2200
2210
2220
n2?:0
2240
2250
2260
2270
2280
2290
2300
2310
2320
2330
2340
2350
2360
URATE
2370
2380
2390
2400
2410
2420
2430
>
*0
2 .J
2460
////////
THEN
"DISC FULL -
INSERT ANOTHER"
"PRESS 'CONTINUE'WHEN
READY"
! //////// ERROR RECOVERY
IF ERRH-64
BEEP
PAUSE
END IF
IF ERRN-54 THEN
BEEP
PRINT "THAT FILE ALREADY EXISTS"
LINPUT "DO YOU WANT TO WRITE OVER THE ORIGINAL DATA CY/N)",Cho$
IF Cho*-"Y"THEN
OFF ERROR
GOTO 1690
ELSE
OFF ERROR
GOTO 1570
END IF
END IF
! ////////////////
LINPUT "DO YOU WANT TO ACQUIRE MORE DATA CY/N)",Cho
IFCho*-"Y"
THEN
PRINT USING".K"
INSERT NEWSAMPLE"
Appendix A
4 of 5
PRIHT
PAUSE
GOTO
END IF
PRINT"FINISHED"
GOTO 2660
i
! ////////
"HIT 'LOCAL', THEN'RESET'
ON DATALOGGER"
"CHECK VOLTAGES FOR LOAD 4 STRAIN AT INITIALSTATE"
ADJUST BALANCE CTLS IFNECESSARY*
"PRESS'CONTINUE'
WHENREADY"
250
////////
Hj'.-> '-
///////
r.etd; ! SUBROUTINE
USES DATA LOGGER BUFFER
60 DATA PTS MAX CASC1I)
! SET FOR SOFTWARE TRIGGER
! CHANGE TO 4 DIGIT DVM
! SPEED DETERMINES TIME BETWEEN RDGS
> ON MICRO-SECONDS)
Had-1
PRINT USING".K"
PRINT "READY FOR Hl-SPEED DATAACQUISITION"
PRINT "BNC JUMPER MUST BE ON DATALOGGER"
PRINT "START INSTRON AND COMPUTER AS SIMULTANEOUSLY AS
RESULTS"
PAUSE
PRINT USING
POSSIBLE FOR ACC
"PRESS'CONTINUE'
WHENREADY-
IF Speed-20
IF Speed=50
I""
Speed1* i li
"#,K"
"DATA ACQUISITION 4 TRANSFER UNDERWAY-PLEASEWAIT"
THEN Tfactor-4000
THEN Tfactor'=2000
"(HO Tftsc tor = 1000
IF Speed =20i THEN Tfactor-'.OO
IF Speed=500 THEN Tfactor=25
2470 OUTPUT 709,"VD4VF1VS1VN60VW"
,Tfactor
2480 OUTPUT 709,'AF15AL16AE1VT3" Appendix A
2490 WAIT FOR DATA ACQ TO END (MAX 30 SEC) 5 of 5
2500 ! RETURN DATA TO COMPUTER AND SORT
2510 OUTPUT 709;"VS"
2520 ENTER 709,Hadata(*)
2530 J-0
2540 PRINTER IS 701
2550 PRINT"
# DISPLACEMENT (MM) LOAD (NEWTONS)
2560 FOR 1-1 TO 59 STEP 2
2570 J-Jl
2580 StralnCJ)-Hadata(I)*Sf
2590 Loadv(J)-Hadata(I*l)>Factorl
2600 IMAGE 3D,5X,90. 40 ,5X ,9D. 4D
2610 PRINT USING 1400 , J,Strain(J) ,Loadv(J)
2620 NEXT I
2630 Speed-Speed(l/60)C1/Gl)100 ( CONVERT TO X/SEC
2640 Speed- ( INTCSpeedxi0000))/ 10000
2650 RETURN
2660 WAIT .5 ///////// END PROGRAM
2670 PRINT USING".K"
2680 OUTPUT 709,"SI"
2690 STOP
2700 END
90
APPENDIX B
COMPUTER PROGRAM
FOR
BRINSON VISCOELASTIC MODEL
91
10 ! VISCOELASTIC MODEL CHARACTERIZATION BY H.F- BRINSON
20 ! PROGRAMMED BY L.T. LOGGI
30 iREQUIRES USE OF BASIC 2.1 AND GRAPHICS 2.1 (MIN)
40 iLAST up0ATE 11_17_87
50 CLEAR 7Appendix B
60 PRINTER IS11 of 5
70 PRINT USING "*,K"5"
BRINSON VISCOELASTICMODEL"
80 PRINT "SOLVES FOR STRESS AS A FUNCTION OF STRAIN FOR TENSILETEST"
90 PRINT USING "3/,K"," VARIABLES:"
100 PRINT"
THETA = LINEAR TO VISCOELASTIC TRANSITION STRESS (kPa)
110 PRINT*
PHI - STRAIN AT TRANSITION POINT(X)"
120 PRINT"
MODULUS - ELASTIC MODULUS(KILO-PASCALS)"
130 PRINT"
(ABOVE 3 VARIABLES DETERMINED FROM ACTUAL TEST DATA)*
140 PRINT
150 PRINT"
TAU - A CORRELATION CONSTANT -- ESTIMATED (SEC)"
160 PRINT"
= NOT THE SYSTEM RELAXATIONTIME"
170 PRINT"
STRAIN - TEST DATA (INDEPENDENT VARIABLE-
X)"
180 PRINT STRESS - TEST DATA (DEPENDENT VARIABLEkPa)'
190 PRINT"
ESTIMATE - CALCULATED STRESS VALUES(kPa)"
200 OPTION BASE 1
210 MASS STORAGE IS"
i INTERNAL ,4 , 0*
220 DIM Streaa(150),StraiTi(150),E3timate(150)
230 INTEGER Record, Rnn
240 DISP "PRESS'CONTINUE'
WHENREADY"
250 PAUSE
260 ON ERROR GOTO 410
270 PRINT USING *#,K"
,"
I NSERT THE DISK WITH STORED DATA INTO THE RIGHT-HAND DR
IVE"
280 PRINT" "
290 PRINT "THIS PGM ASSUMES A PRINTER (ADDRESS 706) IS ON THE HP I BBUS"
300 PRINT" "
310 PRINT "YOU CAN USE THE CRT OR AH EXTERNAL PLOTTER (ADDRESS704)'
320 LINPUT "DO YOU WANT A LISTING OF THE DISK FILES (Y/N)".Choice*
330 IF Choice*[l,lJ="Y"
THEN
340 CAT
350 PRINT
360 END IF
370 PRINT "NAME IS COMBINATION OF STRAIN RATE (IN MM/MIN) AND TEST TYPE (C,
E,G,R)"
380 LINPUT "WHAT IS THE DATA FILE ",File$
390 ASSIGN Pathl TO Fiie$
400 GOTO 600
410 IF ERRN-56 THEN
420 PRINT"
***** THAT FILE IS NOT ON DISC******
430 BEEP 300,.20
440 LINPUT'IS THIS A DUPLICATE/REPEAT TEST (Y OR N)
*
,Repeats
450 IF Repeated, 1]="Y"
THEN
460File$=Fiie*4*D"
470 ELSE
480 PRINT "NAME IS COMBINATION OF STRAIN RATE (IN MM/MIN) AND TEST TYPE CC,
E,S,R)"
490 PRINT 'PLEASERE-SPECIFY"
500 CAT
510 GOTO 380
520 END IF
530 GOTO 320
540 ELSE
550 PRINTER IS 1
560 PRINT ERRM*
570 BZCP 250,. 25
580 PAUSE
590 END IF
92'
600 ! INITIALIZE PARAMETERS
610 Factor-6.990 i KILO-PASCALS PER PSI
620 Ybar-0.
630 Sato-0.
640 Sae-0. Appendix B
650 Sar-0. 2 of 5660 tiae-0.
670 Rmae-0.
680 Ne-0.
690 Rmi-POS(File$,"_") . DETERMINES RUN*
700 IF Rnn-0 THEN
710 Rn*-"0'
720 ELSE
730 Rn$-File$IRnn*lI
740 END IF
750 Ihop-5 i TO SKIP DATA ON PRINTOUT
760 R_aquared-0.
770 ! READ STORED DATA
780 GOSUB Read_data
790 ! CALCULATE ESTIMATED PARAMETERS
800 ! NOTE--PHI AND RATE MUST MATCH UNITS .CURRENTLY USING X STRAIN
810 Modulua-2.34E*6 ! (IN kPa)
820 Theta-2. 10E+4 ! (IN kPa)
830 Phi -.90 ! (IN X)
840 ! PRELIMINARY CALCS
850 GOSUB Find_tau
860 ! NOTE--TAU AND RATE MUST MATCH UNITS FOR CORRECT RESULTS
870 ! CURRENTLY USING SEC ANO 2/SEC
880 Pl-Tau*Modulua*(Rate/100.)
890 P2-Tau*(Rate/100. )
900 ! LOOP CALCULATIONS
910 FOR L-3 TO N
920 P25 = ((Strairi(L)-Phi)/100.)/P2
930 IF P25<0 THEN
940 Eatimate(L)=Modulus*(Straiii(L)/100. )
950 GOTO 1030
960 END IF
970 IF P25>200 THEN P25-200 ! TO AVOID OVERFLOW ERRORS
980 P3-EXPf-P25)
990 P4=i.0-P3
1000 P5-P1*P4
1010 Eatimate(L)=Theta+P5
1020 GOSUB Stat_anal
1030 NEXT L
1040 PRINTER IS 706
1050 PRINT" BRINSON
MODEL"
1060 PRINT
1070 PRINT*
RESULTS FOR STRAIN RATE OF*;Rate;*
Z PERSECOND"
1080 PRINT USING "K ,2D. 2DE ,K,/
*
;*
AND FOR A PREDICT
ED TAU OF";Tau;- SECONDS"
1090 PRINT" C RUN
*;Rn*;' )*
1100 PRINT
1110 PRINT*
STRAIN (X) EXPERIMENTAL STRESS (kPa)
CALCULATED STRESS(kPa)"
1120 FOR L=3 TO 20
1130 IMAGE 5X,4D.2D,15X,8D.2D,21X,8D.2D
1140 PRINT USING 1130 ; St rain(L) ,Stress (L) ,
Eat imate(L)
1150 NEXT L
1160 IF N-30 THEN 'Ihop=l
117.) FIT L=2 TO N STEP inop
1180 RiNF USING 1130 ; Strain(L) ,Str ess (L) ,Est imatt (L.
1190 NEXT L
93
1200
1210
1220
SQUARE
1230
1240
1250
1260
1270
1280
1290
1300
1310
1320
1330
1340
1350
1360
1370
1380
1390
1400
1410
1420
1430
1440
1450
1460
1470
1480
1490
1500
1510
1520
1530
1540
1550
1560
1570
158C
1590
1600
1610
1620
1630
1640
1650
1660
1670
1680
1690
1700
1710
1720
1730
1740
1750
1760
1770
1780
90
1800
1810
ERROR
(kPa)*
IMAGE
USING "1/,X"
LEAST SQUARESANALYSIS:"
ERROR NORMALIZED TO STRAIN (kPa/X) ROOT MEAN
18X,2D.2DE,28X,2D.2DE
PRINT USING 1230,Ne,Rmae
ASSIGN Pathl TO *
PRINT USING !/
PRINT .
PRINTER IS 1
LINPUT "DO YOU WANT TO PLOT THIS DATA (Y/N)",Cho$
IF Cho$tl,l]-"Y*THEN GOSUB Plot_data
LINPUT 'FINISHED ANOTHER TEST FOR ANALYSIS? (Y/N)",Cho
IF Cho$ll,lI-"Y"THEN
PRINT USING"#,K"
GOTO 320
ELSE
GOTO 2890 ! END PGM
END IF
! /////////
! /////////////////
! /////////
Read_data:. ! xxxxxxxx SUBROUTINE READ xxxxxxxxxx
Record-1
PRINT USING"*,K"
ENTER Pathl, Record, N,Rate
EHTER *Pe.thi,2,Gl,Tfactor
PRINT "RATE-";Rate,"GL-*;G1
PRINT "PT# STRAIN STRESS"
FOR M-3 TO N
ENTER Pathl,M;Strain(M) ,Streaa(M)
PRINT M,Strain(M),Streaa(M)
Ybar-Ybar*Streaa(M)
NEXT M
Ybar-Ybar/N
RETURN
Find_tau: ! xxxxxxxxx SUBROUTINE INTERPOLATE xxxxxxxxx
I FACTORS Rl - R5 U'iLL CHANGE W' GAGE LENGTH
IF Gl-5o ."iiEN
Appendix B
3 of 5
Rl
R2 =
R3 =
R4 =
R5 =
.0016
.0166
.1666
1 . 6666
16.6666
.05 MM/MIN CH SPEED
.50 MM/MIN
MM/MIN
MM/MIN
MM/MIN
5.0
50.
500.
END IF
EARLY CALCULATIONS WHEN TRIED TO FIT ALL DATA INDIVIDUALLY
IF Rate>=Rl AND Rate<R2 THEN Tau- ( (Rate-Rl) / (R2-R1) )*(-120 . ) *100 .
IF Rate>=R2 AND Rate<R3 THEN Tau- ((Rate-R2) / (R3-R2) )x(-19. 7) *4 . 2
IF Rate>R3 AND Rate<R4 THEN Tau- ( (Rate-R3) / (R4-R3) )x(-4 . 00) +1 . 0
IF ABS(Rate-R3X1.0E-8 THEN Tau=.95
IF Rate>R4 AND Rate<R5 THEN Tau-( (Rate-R4) / (R5-R4) )x(- . 10) * . 10
IF ABS(Rate-R4Xi. OE-8 THEN Tau-. 18
IF Rate>=R5 THEN Tau=.031
PREDICT TAU AS A LINEAR FUNCTIOH OF STRAIN RATE
CCURRENTLY USING 1 AND 100 MM/MIN AS ENDPTS)
575x(Rate)*2.02Tau = -
RETURN
Stat ana 1 . xxxxxxx SUBROUTINE LEAST SQUARES ANALYSIS xxxxxxx
SSTO - TOTAL SUM OF SQUARES
SSR FEGRLSSJON ?f SHARES
SSE ERROR SUM OF j
NE = ERROR NORMALIZED W/ RESPECT TO STRAIN (IN kPA/X)
9U
I820 ! MSE - MEAN SQUARE ERROR
1830 ! RMSE- ROOT MEAN SQUARE ERROR (IN kPa)
1840 !
1850 ! CHECK FOR BAD DATA AT END OF RECORD
I860 IF StrainCLXStrain(L-l) THEN 1920 Appendix B
1870 ! 4 Of 5
1880 Sato-(Streaa(L)-Ybar)*v2 +Sato
1890 S3e-(Stre33(L)-E3timate(L))*2*Sae
1900 S3r-(Eatimate(L)-Ybar)*l2*Sar
1910 Ne-((Stre33(L)-Eatinate(L))/StraiTi(L))*2 +Ne
1920 IF L-N THEN
1930 R_3quared-S3r/Ssto
1940 Ne-SQR(Ne)/100
1950 M3e-Sse/(N-2)
1960 Rmae-SQR(Mae)
1970 END IF
1980 RETURN
1990 Plot_datas ! xxxxxx SUBROUTINE PLOT xxxxxx
2000 GINIT
2010 GCLEAR
2020 DEG
2030 PRINT "DO YOU WANT TO PLOT ON THE CRT OR EXTERNALPLOTTER"
2040 PRINT 'SELECT C OR P"
2050 PRINT" "
2060 PRINT "(PLOT WILL START IMMEDIATELY AFTER YOU PRESS THE ENTER KEY)"
2070 INPUT PI*
2080 IFP1$-"C"
THEN GOTO 2120
2090 PLOTTER IS704,*HPGL*
2100 PRINTER IS 704
2110 GOTO 2130
2120 PLOTTER IS CRT ,
"
I NTERNAL*
2130 GRAPHICS ON
2140 PEN 1
2150 WINDOW -5,20,-15,50
2160 'CLIP 0,16,0,40
2170 GRID 1,2,0,0
2180 LORG 8
2190 CLIP OFF
22?0 CSIZE 3.5
2^10 FOR \-C TO 4C STEP 4
2220 MOVE 0,1
2230 LABEL I
2240 NEXT I
2250 LDIR 90
2260 LORG 6
2270 MOVE -2,20
2280 LABEL 'STRESS (kPa x10A3)"
2290 LDIR 0
2300 LORG 6
2310 FOR 1=0 TO 16 STEP 2
2320 MOVE 1,0
2330 LABEL I
2340 NEXT I
2350 MOVE 8,-3
2360 LABEL "STRAIN (X)*
2370 CLIP ON
2380 PEN 2
2390 LORG 5
2400 IF N-30 THEN
2410 Iater-1
24>C ELSE
2<30 1st.:, j
2440 END IF
95
2450
2460
2470
2480
2490
2500
2510
2520
2530
2540
2550
2560
2570
2580
2590
2600
2610
2620
2630
2640
2650
2660
2670
2680
2690
2700
271T
2720
2730
2740
2750
CONDS"
2760
2770
2780
2790
2800
2810
2820
2830
2840
2850
2860
2870
2880
2890 !
2900
2910
2920
2930
FOR 1-1 TO 20
IF 1-1 THEN 2480
IF Strain(IXStrain(I-l) THEN 2490
DRAW Strain(I) ,St reaa ( 1 ) / 1000.
NEXT 1
FOR 1-21 TO N STEP latepIF Strain(lXStrain(I-lstep) THEN 2530
Appendix B
5 of 5
Strein(l),Stre33(I)/1000.
I
DRAW
NEXT
PAUSE
PEN 3
CSIZE 1.5
FOR 1-1 TO 16
IF 1=1 THEN 2620
IF Straii-iCIXStrain(I-l) THEN 2620
MOVE Strain(I), Es t iroate( I ) /1000 .
LABEL"X"
NEXT I
FOR 1-17 TO N STEP IstepIF StraindXStraind-Istep) THEN 2670
MOVE Strain(I) ,E3timate(I)/1000.
LABEL"X"
NEXT I
CLIP OFF
PEN 4
CSIZE 3.5
MOVE 8,45
LACEL USiK:*
CSIZE 2.0
MOVE 8,42
LABEL USING
"K,2A,K"
;*POLYSTYRENE- TENSILE - RUN ";Rn*
'K, 2D. 2DE,K"
; "BRINSON MODEL W/ PREDICTED TAU OF ";Tau;"
SE
CSIZE 2.0
MOVE 4,-10
LABEL "TEST SPEED:-,Rate;"
X PERSECOND*
IFFile*tl,lJ="E*
THENTran-"EXTENSOMETER*
PR
PR
WA
PR
IFFile$tl,lI-"S*
THEN Tran*>
IFFile[l,ll="C"
THEN Tran-
MOVE 3.3,-12
LA8CL "TRANSOUCER: *;Tran
MOVE 12,-10
LABEL "xxTHEORETICAL"
LABEL "--EXPERIMENTAL-
PRINTER IS 1
RETURN
END op PROGRAM
INT USING*#,K"
INT "THAT'S ALL FOLKS!"
IT .05
INT USING"#,K"
STRAINGAGE*
CROSSHEADDISP"
2940 STOP
2950 END
APPENDIX C
COMPUTER PROGRAM
FOR
MATSOUKA VISCOELASTIC MODEL
97
10
20
30
40
50 !
60
70
80
90
TEST
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
IVE"
340
350
360
370
380
390
400
410
420
430
440
450
460
470 I
480
490
500
510
520
530
540
E,S,R)
550
560
57058'
590
600
VISCOELASTIC MODEL BY S. MATSOUKA
PROGRAMMED BY L.T. LOGGI
MODIFIED 5/87 FOR BASIC 3.0 AND GRAPHICS 3.0
--LAST UPDATE 03-07-88
CLEAR 7
OPTION BASE 1
PRINTER IS 1
PRINT USING "#,K",*
MATSOUKA VISCOELASTICMODEL'
PRINT "SOLVES FOR STRESS AS A FUNCTION OF STRAIN FOR TENSILE OR RELAXATION
Appendix
1 of
EPSILON ZERO
EPSILON BAR
MODULUS
TAU
BETA
STRAIN
STRESS
ESTIMATE
VARIABLES:"
STRAIN AT MAX STRESS (YIELDPOINT)'
KNOWN FROM TEST DATA(X)'
STRAIN VALUE AT ANY TIME-f
READ FROM DATA FILE(X)'
RELAXATION MODULUS SLOPETERM"
- KNOWN VALUE(EMPIRICAL)'
A CORRELATION CONSTANT -- ESTIMATED(SEC)'
NOT EQUAL TO THE RELAXATIONTIME'
EXPONENTIAL FACTOR, A MATERIALCONSTANT"
KNOWN VALUE(EMPIRICAL)"
TEST DATA (INDEPENDENT VARIABLE -X)"
TEST DATA (DEPENDENT VARIABLE - kPa)"
CALCULATED STRESS VALUES(kPa)"
WHENREADY'
PRINT USING "1/"
PRINT "
PRINT'
PRINT"
PRINT"
PRINT"
PRINT"
PRINT"
PRINT"
PRINT"
PRINT"
PRINT*
PRINT"
PRINT"
PRINT*
DISP "PRESS -continu
PAUSE
DIM Ep3ilon(150) ,Strain(150) ,Streaa(150) ,Eatiroate(150)
REAL Initial, Integrand
INTEGER Record, Rnn
i FILE MANIPULATION.
MASS STORAGE IS ": INTERNAL ,4,0
*
ON ERROR GOTO 470
PRINT USING*e,K',"
INSERT THE DISK WITH STORED DATA INTO THE RIGHT-HAND DR
PRINT "THIS PGM ASSUMES A PRINTER (ADDRESS 706) IS ON THE HPIBBUS-
PRINT "YOU CAN USE THE CRT OR AN EXTERNAL PLOTTER (ADDRESS~?04)"
LINPUT "DO YUU WANT A LISTING OF FILES I'N J1SK f.Y/N)*,ChoS
IFCho*tl,lI-'Y"
THEN
CAT
PRIHT
END IF
PRINT "NAME IS COMBINATION OF STRAIN RATE (MM/MIN)
LINPUT "WHAT IS THE DATA FILE ",File
ASSIGN GPathl TO File*
GOTO 650
IF ERRN-56 THEN
PRINT USING"K,/*;-
xxxxxx
BEEP 300,-20
LINPUT"
IS THIS A DUPLICATE/REPEAT TEST (Y OR N)
IFRepeetSll.il-'Y*
THEN
File*=File*4"D"
ELSE
PRINT 'NAME IS COMBINATION OF STRAIN RATE (MM/MIN)
PRINT "PLEASERE-SPECIFY"
CAT
GOTO 440
END IF
GOTO4C
..>
ELSE
4 TEST TYPECC,E,G,R)'
THAT FILE IS NOT ON DISC xxxxxx
,Repeat*
4 TEST TYPE (C,
98
610
620
630
640 END IF
650 ! ---
660
670
680
690
700
710
720
730
740
750
760
770
780
790
800
810
820
830
840
850
860
870
880
890
900
910
920
930
940
950
960
970
980
990
PRINT ERRM*
BEEP 250,. 25
PAUSE
Faclor-6. 990
Ybar-0
Sato-O.
Sae-0.
Sar-0.
Mae-O.
Rmae-O.
Ne-O.
R_aquared-0.
Rnn-POS(File, "_')
IF Rnn-0 THEN
Rn*-"0*
ELSE
Rn*-File*[Rnn+l J
END IF
lhop-3
Naect ion-ID
i
! ,.
Beta-. 0242
Modulu3-(5.85E*5)XFactor
INITIALIZE PARAMETERS
! KILO-PASCALS PER PS1
Appendix C
2 of 6
DETERMINES RUN*
! TO SKIP DATA ON PRINTOUT
-ESTABLISH EMPIRICAL CONSTANTS-
! (IN kPa)
iUSE TEST DATA T0 ESTABLISH SOME MODEL PARAMETERS-
Ep3ilon_zero-1. 5/100 ( (IN X STRAIN)
i
READ STORED DATA
GOSUB Read data
PLEASE WAIT WHILE THE CALCULATIONS ARE BEINGMADE"
- LOOP CALCULATIONS
! CALCULATE ESTIMATED PARAMETERS
! NOTE TAU AND RATE MUST MATCH UNITS FOR CORRECT RESULTS
! CURRENTLY USING SEC AND X/SEC
GOSUB Find_tau
Constant-TauX(Rate/100) ! CONVERT FROM X
PRINT USING,?"
it: a prikt-
1010 PRINT
1020 !
1030 !
1040 FOR L-3 TO N
1050 Ep3ilon_bar-Strain(L)/100.
1060 GOSUB Integration
1070 E3timate(L)-(Delta_epailon/3)x(Initial*2.xSuodd*4.xSui-even+Final)
1080 GOTO 1100
1090 PRINT USING "K ,5D. 2D ;*
after the integration subroutine: STRE
SS ="
5Eatimate(L)
1100 GOSUB Stat_anal
1110 NEXT L
1120 PRINTER IS 706
1130 PRINT"
MATSOUKAMODEL*
1140 PRINT"
RESULTS FOR STRAIN RATE OF";Rate;"
X PER SECON
D"
1150 PRINT USING "K,2D. 2DE -,
"
AND FOR AN PREDICTED TAU OF
";Tau
1160 PRINT*
( RUN-;Rn*;- )"
1170 PRINT USING / ,i';*
STRAIN (X) EXPERIMENTAL STRESS (kPa) CA
LCULATED STRESSCkPeO*
1180 FOR L=3 i0 2 0
1190 IMAGE 3X,4D.2D,13X,8D.2D,24X,8D.2D
99
1200
1210
1220
1230
1240
1250
1260
1270
1280
ERROR
1290
1300
1310
1320
1330
1340
1350
1360
1370
1380
1390
1400
1410
1420
1430
1440
1450
1460
1470
1480
1490
1500
1510
1520
1530
1540
1550
1560
1570
1500
1590
1600
1610
USING 1190;Strain(L),Stre33(L),Eatimate(L)
LEAST SQUARES
ERROR NORMALIZED TO STRAIN (kPa/X)
THIS DATA (Y/N)*,Cho<
Piot_data
(Y/N)*,Cho*
NEXT L
IF N-30 THEN Ihop-1
FOR L-21 TO N STEP IhopPRINT USING 1190,Strain(L),Streaa(L),E3tiate(L)
NEXT L
PRINT USING "1/*
(kPa)*
IMAGE 21X,2D.2DE,28X,2D.2DEPRINT USING 1290,Ne,Rp,ae
ASSIGN Pathl TO X
PRINT USING '1/"
PRINTER IS 1
PRINT USING",K"
LINPUT 'DO YOU WANT TO PLOT
IF Cho*[l,ll-'Y*THEN GOSUB
LINPUT 'FINISHEDANOTHER TEST FOR ANALYSIS?
IF Cho*Il,ll-"Y"THEN
PRINT USINGm9,K*
GOTO 330
ELSE
goto 3310 ! end pgm
e^.:> if
! //'
/ ,///////
! ////////////////////
! ///////////
Read_data: ! xxxxxxxxxxxxxx SUBROUTINE- --
Record-1
PRINT USING*,K"
ENTER Pathl,Record,N,Rate
ENTER Pathl,2,Gl,Tfactor
PRINT'
RATE-",Rate;"
GL-";G1
PRINT "PT# STRAINSTRESS"
FOR M-3 TO N
EHTER *Pithl,M;Strain(M) ,StreasCM)
PRINT h,Strain(M) ,Stre33(M)
Ybar-Ybar*Streas(M)
NEXT M
Ybar-Ybar/N
Appendix C
3 of 6
ANALYSIS:"
ROOT MEAN SQUARE
READ XXXXXXXXKXXXXXXXXX
1620 RETURN
1630 Find t-
1640
1650
1660
1670
1680
1690
1700
1710
1720
1730
1740
1750
1760
1770
1780
4"*
-,
J .
1 '. jO
1810
u >
Tau
RETURN
I ntegrat ion
PRINTER IS
! xxxxxxxxxxxxxx SUBROUTINE --
ESTIMATE
(1.51E*5xRate)-2520.
XXXXXXKXXXXXXXXX
xxxxxxxxxxxxxx SUBROUTINE INTEGRATE IxJiXXXXXXXSiitlX
706
SIZE
FOR ESTIMATION -- MUST BE EVEN #
> USING SIMPSON'S RULE
! DELTA_EPSILON-STEP
NSECTION-* OF SECTIONS
Del ta_epsi lon-Epsi lon_bar/N3ect ion
IF Epsi lon_bar>Epai lon_zero THEN
Naection-20
Del ta_epai lon-Epai lon_bar/Nsect ion
END IF
Sumeven-0. ! INITIALIZE THESE PARAMETERS FOR EACH INTEGRATION
Sumodd=0.
Initial=0.
F jr,al = C
1 ntegrand-0 .
GOTO 1840 100
PRINT "DELTA E -"
;Del ta_epai Ion
PRINT USING "K";"
J EPSILON(J) I NTEGRAND(J)"
FIND VALUE OF INTEGRAND FOR EVERY EPSILON FROM 1 TO N*l
! KNOW THAT EPSILON(l-l) = 0 Appendix C
! AND THAT EPSILOH( I -N*l) - EPSILON_BAR 4 of 6
Epailon(l>-0.
Initial-FNCelcd ,Epailon(x) ,Epai lon_zero ,Epai lon_bar .Modulus
,Beta .Constant
1820
1830
1840
1850
1860
1870
1880
)
1890 FOR 1-2 TO Nsection STEP 2
1900 Ep3ilonCI)-(I-l)xDelta_epailon
1910 ! NOW CALCULATE THE COMPONENTS OF THE FUNCTION (INTEGRAND)
1920 EVEN VALUES FOUND FOR i-2 TO i-N
1930 Number 1-FNCa led .Epailon(x) ,Ep3ilon_zero,psi lon_bar .Modulus ,Beta ,
Constant
)
1940 Sumeven-Su-aeven+Nu-aberl
1950 IF ALL SECTION CALCS HAVE BEEN DONE, THEN FIND INTEGRAND
I960 FOR EPSILON AT i-N*l
1970 IF I-Naection THEN
1980 Epsi Ion(Naect ion*1) -Epai lon_bar
1990 Final-FNCalc(Naect ion*l, Epai lon(x) ,Epailon_zero ,
Epai lon_bar .Modulus ,B
eta,Constant)
2000
2010
2020
2030
2040
2050
nt)
2060
GOTO 2070
END IF
! NOW EVALUATE FUNCTION FOR THE NEXT EPSILON(i)
! ODD VALUES FOUND FOR i=3 TO i-N-1 ONLY
Epailon(I*l)-(I*l-l)XDeita_epailon
Nunber2-FNCalc( I+l,Epailon(x) ,psi lon_zero .Epsi lon_bar .Modulus .Beta ,Cons ta
Sumodd=Sumodd+Number2
2070 NEXT I
2080 GOTO 2120
2090 IMAGE 2D. 2DE.3X ,20. 2DE ,8X ,D. D, 15X ,Z. 2D,9X ,2D. 2DE ,7X ,2D. 2DE ,
/
2100 PRINT USING '/ , "SUMODD SUMEVEN IN1T EPSILON FINAL EPSILO
N INIT INTEG FINALINTEG"
2110 PRINT USING 2090 ;Sumodd,Sumeven,Epai lon(l) ,Epai lon(Naect ion*l) ,
I nit ial ,Fin
al
2120 RETURN
2130 Stat_anal>
2140
2150
2160
2170
2180
2190
2200
2210
2220
2230
2240
2250
2260
2270
2280
2290
2300
2310
xxxxxxxxxx SUBROUT 1 NE LEAST SQUARES ANALYSIS xxxxxxx
TOTAL SUM OF SQUARES
REGRESSION SUM OF SQUARES
ERROR SUM OF SQUARES
ERROR NORMALIZED W/ RESPECT TO STRAIN (IN kPa/X)
MEAN SQUARE ERROR
ROOT MEAN SQUARE ERROR (IN kPa)
SCTC
5SR
SSE
NE
MSE
RMSE
IF Strain(LXStrain(L-l) THEN 2250
Sato-(Stre3s(L)-Ybar)"2*Ssto
S3e-(Stre3a(L)-Eatimate(L))-2*Sae
S3r-(Estimate(L)-Ybar)*2*Ssr
Ne=((Stre3s(L)-Estimate(L))/Strain(L))*>2*Ne
IF L =N OR Strain(LXStrain(L-l) THEN
R_aquared=Sar /Sato
Ne-SQR(Ne)/100
Mse=Sse/(N-2)
Rmse-SQR(Mse)
END IF
RETURN
2320 Plot_data:
2330
2340
2350
2360
KXXKXXKX SUBROUTINE PLOT xxxxxxx
2370
2380
GINIT
GCLEAR
DEG
ALPHA OFF
PR INI "1.0
PRINT "SELECT
YOL-ff>N' Ui PLOT ON Th CiTF OR ANOTHER
PLOTTER1
ORP*
101
2390
2400
KEY)'
2410
2420
2430
2440
2450
2460
2470
2480
2490
2500
2510
2520
2530
2540
2550
2560
2570
2580
2590
2600
2610
2620
2630
2640
2650
2660
2670
2680
2690
2700
2710
2720
2730
2740
2750
2760
2770
2780
2790
2800
2810
2820
2830
2840
2850
2860
2870
2880
2890
290 0
2910
2920
2930
2940
2950
2960
2970
2<0
LJ90
3000
PRINT""
PRINT "(PLOT WILL START IMMEDIATELY AFTER YOU PRESS THE RETURN/ENTER
INPUT PI*
IF Pl*--C"THEN GOTO 2500
IF P1<>"C AND P1<>"P"THEN
BEEP
GOTO 2370
END IF
PLOTTER IS 704,'HPGL"
PRINTER IS 704
GOTO 2510
PLOTTER IS CRT,"INTERNAL"
GRAPHICS ON
PEN 1
WINDOW -5,20,-15,50
CLIP 0,16,0,40
GRID 1,2,0,0
LORG 8
CLIP OFF
CSIZE 3.5
FOR 1-0 TO 40 STEP 4
MOVE 0,1
LABEL I
NEXT I
LDIR 90
LORG 6
MOVE -2,18
LABEL "STRESS (kPa x10*3)"
LDIR 0
LORG 6
FOR 1-0 TO 16 STEP 2
MOVE I, 0
LABEL I
NEXT 1
MOVE 8,-3
LABEL "STRAIN(X)"
CLIP ON
PEN 2
LORG 5
IF N-30 THEN
Iatep=l
ELSE
Istep-3
END IF
FOR 1=1 TO 20
IF 1-1 THEN 2860
IF StraindXStrain(I-l) THEN 2870
DRAW Strain(I) .Stress ( I ) /1000
NEXT I
FOR 1=21 TO N STEP Istep
IF 1=1 THEN 2910
IF StraindXStrain(I-l) THEN 2920
DRAW Strain(I),Stress(I)/1000
NEXT I
PAUSE
PEN 3
CSIZE 1.5
FOR 1=1 TO 20
IF 1=1 THEN 2990
5tr=iin(IX?tr,jind 1) THEN 3010
rjVE Straind) ,Est ima led ) / 1000
LABEL-?'
102
Appendix C
5 of 6
3010
3020
3030
3040
3050
3060
3070
3080
3090
3100
3110
3120
3130
3140
3150
3160
3170
3180
3190
3200
3210
3220
3230
3240
3250
3260
3270
3280
3290
3300
3310
3320
3330
3340
3350
3360
3370
3380
3390
3^00
3410
3420
3430
3440
3450
3460
3470
3480
3490
Appendix C
6 of 6
TENSILE - RUN ";Rn*
i Tau
NEXT I
FOR 1-21 TO N STEP latepIF 1-1 THEN 3050
IF Strain(IXStraind-l) THEN 3070
MOVE Strain(I),E3timate(I)/1000
LABEL*?-
NEXT I
CLIP OFF
PEN 4
CSIZE 3.5
MOVE 8,45
LABEL USING *K ,2A , "POLYSTYRENE-
CSIZE 2.0
MOVE 8.1,42
LABEL USING 'K, 2D.2DE"
, 'MATSOUKA MODEL W/ PREDICTED TAU OF
CSIZE 2.0
MOVE 4,-10
LABEL "TEST SPEED.-,Rate,"
X PERSECOND"
IFFile*ll,lJ-"E"
THEN Tran*- -EXTENSOMETER"
IFFile*Cl,ll-"S"
THEN Tran*- 'STRAIN GAGE"
IFFile*Il,lJ-"C"
THEN Tran- "CROSSHEADDISP"
MOVE 3.4,-12
LABEL "TRANSDUCER: *,Tran*
MOVE 12,-10
LABEL "?? THEORETICAL"
LABEL' EXPERIMENTAL-
PRINTER IS 1
WAIT 3
GCLEAR
RETURN
i EH0 0F PROGRAM.
PRINT USING-e.K'
PRINT "FINISHED, FINALLY!>"
WAIT .25
PRINT USING"#,K"
STOP
END
! xxx -- ROUTINE TO CALCULATE THE FUNCTION OF EPSILON -- xxx
DEF FJ-!C?ilc (J>pt.-i i orCw) .Epai 1 on_zero ,Eps i lon_bar .Modu lus .Beta,Constant)
Part:1KP(-Epai'
ar.C'./^pai lon_zero)
Par t2-Epsil on_bar -Epsi Ion (J)
Par t3-Part2/Constant
Part4-EXP(-(Part3*Beta))
Integrand-ModulusXPortlxPart4
GOTO 3480
IMAGE 2D,12X,Z.3D,22X,7D.2D
PRINT USING 3460;J,Epsilon(J) ,Integrand
RETURN integrand
FNEND
