CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
2002 American Institute of Physics 0-7354-0068-7
THERMAL ACTIVATION CONSTITUTIVE MODEL FOR POLYMERS
APPLIED TO POLYTETRAFLUOROETHYLENE
Frank J. Zerilli and Ronald W. Armstrong*
Research and Technology Department, Naval Surface Warfare Center Indian Head Division, Indian Head, MD
20640-5035
Abstract. A constitutive model with a thermal activation plastic component combined with a
Maxwell-Weichert linear viscoelastic component was applied to describe the dynamic elastic and
plastic stress-strain behavior of polytetrafluoroethylene as a function of temperature, strain rate, and
pressure. The linear viscoelastic component that describes the small strain behavior is given an
explicit temperature dependence through Arrhenius-like temperature dependent relaxation times. The
Arrhenius activation energies also include a linear pressure dependence. In the plastic component, the
shear activation volume is taken to be inversely proportional to the shear stress and both the shear and
dilatational activation volumes are functions of pressure. Strain hardening in the plastic component
is described through the creation and destruction of flow units as defined by Kauzmann.
INTRODUCTION
crystalline polytetrafluoroethylene (PTFE) as
functions of temperature, pressure, and strain rate.
The PTFE in this study is assumed to have a density
of 2.15 g cm"3 which would fix the degree of
crystallinity in the range of 55-60 %, depending on
the concentration of micro voids.
We describe an equation which addresses the
temperature, strain-rate, and pressure dependence of
a stress-strain relation that is applicable to polymers.
Brittleness and cracking associated with strain-rate,
temperature, and pressure effects are not addressed.
It is assumed that the strain-rates and temperatures are
in the range in which creep and long-term relaxation
effects are negligible. Experimental evidence
indicates that most, if not all, of the apparent plastic
deformation is recoverable, given enough time or
high enough temperature. This recovery is not treated
here. The model is three dimensional but isotropic.
The total strain is divided into a viscoplastic part
and a viscoelastic part. The viscoplastic part is
described by a non-linear thermal activation dashpot
similar to that described by Eyring (1), and the
viscoelastic part is described in terms of a MaxwellWeichert linear viscoelasticity (2).
The equations are shown to describe the major
features of the reported stress-strain curves of semi-
POLYMER CONSTITUTIVE EQUATION:
VISCOPLASTIC COMPONENT
The thermal activation model was described
previously (3). It is assumed that the materials are
isotropic, that there is no dependence on the third
stress invariant, that the pressure volume of activation
is a function of pressure only, and that the effective
stress activation volume depends inversely on the
effective stress.
The strain hardening (4) is described abstractly in
terms of units offlow as defined by Kauzmann (5). In
Kauzmann's definition, units of flow are the
"structures in a body whose motions past one another
make up the unit shear stress process ... ".
'Present Address: Air Force Research Laboratory, Eglin AFB, Florida 32542-5910
657
TABLE 2. Maxwell-Weichert parameters for the initial
linear viscoelastic deformation of PTFE.
TABLE 1. Parameters for the viscoplastic deformation of
PTFE.
1
£0pa(MPa)
£<)Pb
2.01 x lO"2
2.64 x lO'4
4.78 x lO'3
5.02 x lO-5
-8.2
-0.625
-7.0 x 10'3
4016
2.0x1 0'2
0.714
72.4
2.2x1 0-2
A)pn
0.5
Po(K - )
1
P i ( K- )
-1)
1
Hi ^fV - )
_,
fir
cob
WpCMPa 1 )
£pa(MPa)
Bpb
#pn
k
1
2
3
4
5
6
7
8
ii
of
//
2q| 2G2|
~J
p
'
„
f
B = B er(1 + B b^'
.p)
Qpn
o'^
(4)
POLYMER CONSTITUTIVE EQUATION:
VISCOELASTIC COMPONENT
A Maxwell-Weichert model is used to describe the
initial viscoelastic stress. This model is placed in
series with the non-linear dashpot described by Eq.
(1) above, as illustrated in Fig. 1. The total
deformation rate is divided into viscoelastic and
viscoplastic parts, and the viscoelastic deformation
rate has an elastic and a viscous component.
The equations for the stress deviators may be
written down by inspection of Fig. 1 and are not
reproduced here. For many polymers, the bulk
response is nonlinearly elastic with negligible
viscosity. In this case, the trace of the total stress and
deformation rates are related by
an - 3A:(e..)e..
v(5)
^ ir ii
'
Following Bergstrom's analysis for dislocations in
iron (6), the total flow unit density may be related to
the plastic strain by a differential equation describing
the mobilization and immobilization of flow units.
The resulting viscoplastic component of the
deformation is described by the equation
a = Be~^T + Bj(l- e'^/co e~aT
n)
v /
where the quantities Bpa, Bpb, Bpn, B0pa, B0pb, B0pn are
constants.
FIGURE 1. Maxwell-Weichert linear visco-elasticity plus thermal
activation non-linear viscoplasticity.
(1)
where
P = P0 -
4.2x1 0'6
#<>k(K)
2338
10740
9640
5920
22920
1160
2320
1106
where coa, cob, cop are constants. The pressure
dependence for the coefficients B and B0 is
|f)(e)|(e).
\ fT
"T5S r> *
a = aQ -
2.7x1 0'5
2.1xlO'5
co = coa + cojne
+ cojrp
o
a
2Gn|
*Yr
T()k (sec)
1.7xlO'7
2.2xlO'17
5.1xiO'12
4.6xlO'9
4.1xlO'18
for PTFE, co, the flow unit mobilization rate, depends
approximately linearly on the pressure p and logarithmically on the strain rate 8, so we choose
G
^
Gk (MPa)
950
350
90
100
40
200
200
200
(2)
and, neglecting their potential pressure dependence,
po, P1? a0, d] are constants. From the experimental data
658
PTFE, 64% CRYSTALLINITY
1000
CO
Q 100
o
s
.V-
10
Data, 1 Hz, McCrum
Maxwell-Weichert Model
0.0
200 300 400 500 600
TEMPERATURE (K)
FIGURE 2. Fit of Maxwell-Weichert model with eight
components to storage shear modulus data for 64% crystallinity
PTFE at 1 Hz (Ref. 13). The data cover the temperature range from
4.2 K to the melting point at 600 K.
0
100
0.1
0.2 0.3 0.4
0.5
TRUE TOTAL STRAIN
0.6
FIGURE 3. Total stress-strain calculated with viscoelastic/plastic
model and the parameters in Tables 1 and 2, compared with split
Hopkinson pressure bar data reported by Walley and Field (Ref.
10) for PTFE.
where K is a volume and temperature dependent bulk
modulus.
The relaxation times, ik = \/2Gk, have a
temperature and pressure dependence given by
(6)
where the activation energies (in units of temperature)
are
H
k
= H
ok
+ A
PkP
'
(?)
APPLICATION TO
POLYTETRAFLUOROETHYLENE
0.00 0.04 0.08 0.12 0.16 0.20 0.24
TRUE TOTAL STRAIN
PTFE has an unusually complicated phase
diagram with four crystalline forms and a liquid
phase (7). The amorphous part of the material has
several glass-like transition temperatures, designated
a, p, and y by McCrum (8). It also exhibits significant
non linear viscoelastic behavior down to small strains
of the order of 1% (9). Most of the experimental
constitutive data for PTFE used for the analysis in
this work lies in the temperature and pressure range
in which the crystal is triclinic and the amorphous
material is above the y transition and below the p
transition.
The parameters for use in the constitutive equation
were obtained by analyzing results reported by a
number of investigators: compressive split Hopkinson
FIGURE 4. Total stress-strain calculated with viscoelastic/plastic
model and the parameters in Tables 1 and 2, compared with split
Hopkinson pressure bar data reported by Gray (Ref. 11) for PTFE.
pressure bar stress-strain curves at a number of strainrates reported by Walley and Field (10), Hopkinson
bar data at a number of temperatures and strain-rates
determined by Gray (11), and tensile stress-strain data
at various superposed hydrostatic pressures reported
by Sauer and Pae (12). The parameters derived from
these data are given in Table 1.
An eight component Maxwell-Weichert model
was used for the viscoelastic part, with the moduli,
relaxation times, and activation energies,
G, T , H , respectively, chosen to give a
659
activation volume for thermally activated flow. The
upward curvature of the strain hardening curve is
accounted for by a pressure and strain rate dependent
rate of immobilization of flow units. The model does
not describe long term relaxation or creep effects.
Due to the inclusion of pressure dependence, it is
expected to predict the difference in stress between
compressive and tensile loading.
ACKNOWLEDGMENTS
0.0
0,2 0.4 0.6 0.8 1.0
TRUE TOTAL STRAIN
This work was principally supported by the
NSWC In-House Laboratory Independent Research
Program with partial support from the Office of
Naval Research. Additional partial support was
provided by NSWC for Ronald Armstrong.
Appreciation is also expressed to G. T. Gray, III, for
providing us his Hopkinson bar data on PTFE in
advance of publication.
1.2
FIGURE 5. Total stress-strain calculated with viscoelastic/plastic
model and the parameters in Tables 1 and 2, for different
pressures, compared with tensile test data reported by Sauer and
Pae (Ref. 12).
reasonably good match to the shear storage modulus
and logarithmic decrement vs. temperature data at 1
Hz published by McCrum (13). As illustrated in Fig.
2 for the storage modulus, the match is good except
at the highest temperatures near the melting point.
For the purpose of the comparisons presented
here, it was assumed that PTFE is incompressible, so
that the effective elastic (Young's) moduli, Ek, are
three times the shear moduli, Gk. This is a good
approximation because PTFE's bulk modulus is
significantly larger than its shear modulus.
The viscoelastic pressure volume of activation A
REFERENCES
1. Eyring, H.,7. Chem. Phys. 3,107 (1935);4,283 (1936).
2. Tschoegl, N. W., "The Phenomenological Theory of
Linear Viscoelastic Behavior", Springer-Verlag, Berlin,
1989, p. 158.
3. Zerilli, F. J., and Armstrong, R. W., in Shock
Compression of Condensed Matter — 1999, edited by
M. D. Furnish, L. C. Chhabildas, and R. S. Hixon, AIP
Conference Proceedings 505, New York, 2000, pp. 531
ff
4. Zerilli, F. J. and Armstrong, R. W., J. Phys. IVFrance
10, 3 (2000).
5. Kauzmann, W., Trans. Am. Inst. Min. Metall. Eng. 143,
57(1941).
6. Bergstrom, Y., Mater. Sci. and Eng. 5,193-200 (1970).
7. Flack, H. D.,J. Polymer Science A-2 W, 1799(1972).
8. McCrum, N. G., Read, B. E., and Williams, G.,
Anelastic and Dielectric Effects in Polymeric Solids,
Wiley, New York, 1967.
9. Thomas, D. A., Polymer Eng. and Sci. 9, 415 (1969).
10. Walley, S. M., and Field, J. E., DYMAT Journal 1,211227, Fig. 2o (1994).
11. Gray, G. T., Ill, in Proc. Plasticity '99, Seventh Int.
Symp. on Plasticity and Its Current Applications, edited
by A. S. Khan, Neat Press, Fulton, Maryland, 1998.
12. Sauer, J. A., and Pae, K. D., Colloid & Polymer Sci.
252,680(1974).
13. McCrum, N. G., J. Polymer Sci. 34, 355 (1959).
was assumed to have the same value for each
Maxwell element. It was set at a value of 4.0 K/MPa
so that the calculated initial slope of the stress-strain
curve would match that of the data of Sauer and Pae.
Figures 3 - 5 show the complete stress-strain
curves calculated with the parameters in Tables 1 and
2 compared to the reported experimental data.
CONCLUSION
A constitutive equation has been developed that
describes reasonably well the effective stress for
viscoplastic flow in PTFE in uniaxial dynamic
tension tests as a function of strain, strain rate,
temperature. A key element of this analysis is the
assumption of the inverse stress dependence of the
660