We follow the assumption that the dilatational response of glassy polymers can be characterized by a back-stress type analog that includes a thermal expansion for each elastic component and with a viscosity that is dependent on the expansion of the elastic back-stress component. To this, we add the assumption of an unloaded equilibrium temperature that correlates to the past processing through the viscous flow. After setting this in a thermodynamically consistent structure, elastic, elastic back-stress, thermal expansion, back-stress thermal expansion, heat capacity, and viscous damping are evaluated using existing experiments for the response of polycarbonate over the glassy and rubbery ranges. For the demonstration, this is done entirely using a WLF shift factor that is augmented to include, in addition, back strain superposition. We then examine the resulting model under different thermal and mechanical loadings that have the material passing through the glass transition.

1.
Tool
,
A. Q.
, and
C. G.
Eicitlin
, “
Variations caused in the heating curves of glass by heat treatment
,”
J. Am. Ceram. Soc.
14
(
4
),
276
308
(
1931
).
2.
Hutchinson
,
J. M.
, “
Physical aging of polymers
,”
Prog. Polym. Sci.
20
(
4
),
703
760
(
1995
).
3.
Cangialosi
,
D.
,
V. M.
Boucher
,
A.
Alegría
, and
J.
Colmenero
, “
Physical aging in polymers and polymer nanocomposites: Recent results and open questions
,”
Soft Matter
9
(
36
),
8619
8630
(
2013
).
4.
Merrick
,
M. M.
,
R.
Sujanani
, and
B. D.
Freeman
, “
Glassy polymers: Historical findings, membrane applications, and unresolved questions regarding physical aging
,”
Polymers
211
,
123176
(
2020
).
5.
Struik
,
L. C. E.
, “
Volume relaxation in polymers
,”
Rheol. Acta
5
(
4
),
303
311
(
1966
).
6.
Struik
,
L. C. E.
,
Physical aging in amorphous polymers and other materials, Ph.D. thesis, Delft University of Technology, Delft, Netherlands
,
1977
;
available at
https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=965f3a3d3f8a9960846bcd9e91d4ae55f6cc a44a.
7.
Kovacs
,
A. J.
, “
Transition vitreuse dans les polymères amorphes. Etude phénoménologique
,” in
Fortschritte der hochpolymeren-forschung
(
Springer
,
New York
,
1964
), pp.
394
507
.
8.
Kovacs
,
A. J.
, “
La contraction isotherme du volume des polymères amorphes
,”
J. Polym Sci.
30
(
121
),
131
147
(
1958
).
9.
Zheng
,
Q.
,
Y.
Zhang
,
M.
Montazerian
,
O.
Gulbiten
,
J. C.
Mauro
,
E. D.
Zanotto
, and
Y.
Yue
, “
Understanding glass through differential scanning calorimetry
,”
Chem. Rev.
119
(
13
),
7848
7939
(
2019
).
10.
Tropin
,
T. V.
,
J. W.
Schmelzer
, and
V. L.
Aksenov
, “
Modern aspects of the kinetic theory of glass transition
,”
Phys. Usp.
59
(
1
),
42
66
(
2016
).
11.
Simon
,
S. L.
, and
G. B.
McKenna
,
Structural recovery and physical aging of polymeric glasses
, in
Polymer Glasses
(
CRC
, Boca Raton,
2016
), pp.
39
70
.
12.
Caruthers
,
J. M.
, and
G. A.
Medvedev
,
Thermo-mechanical signatures of polymeric glasses
, in
Polymer Glasses
(
CRC
, Boca Raton,
2016
), pp.
123
194
.
13.
Medvedev
,
G. A.
, and
J. M.
Caruthers
,
A comparison of constitutive descriptions of the thermo-mechanical behavior of polymeric glasses
, in
Polymer Glasses
(
CRC
, Boca Raton,
2016
), pp.
467
552
.
14.
Lion
,
A.
,
C.
Mittermeier
, and
M.
Johlitz
, “
Heat capacities and volumetric changes in the glass transition range: A constitutive approach based on the standard linear solid
,”
Contin. Mech. Thermodyn.
29
(
5
),
1061
1079
(
2017
).
15.
Negahban
,
M.
,
The Mechanical and Thermodynamical Theory of Plasticity
(
CRC
,
New York
,
2012
).
16.
Goldstein
,
M.
, “
Some thermodynamic aspects of the glass transition: Free volume, entropy, and enthalpy theories
,”
J. Chem. Phys.
39
(
12
),
3369
3374
(
1963
).
17.
Badrinarayanan
,
P.
,
W.
Zheng
,
Q.
Li
, and
S. L.
Simon
, “
The glass transition temperature versus the fictive temperature
,”
J. Non-Cryst. Solids
353
(
26
),
2603
2612
(
2007
).
18.
Williams
,
M. L.
,
R. F.
Landel
, and
J. D.
Ferry
, “
The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids
,”
J. Am. Chem. Soc.
77
(
14
),
3701
3707
(
1955
).
19.
Ravi-Chandar
,
K.
, and
Z.
Ma
, “
Inelastic deformation in polymers under multiaxial compression
,”
Mech. Time Depend. Mater.
4
(
4
),
333
357
(
2000
).
20.
Kovacs
,
A. J.
,
J. J.
Aklonis
,
J. M.
Hutchinson
, and
A. R.
Ramos
, “
Isobaric volume and enthalpy recovery of glasses. II. A transparent multiparameter theory
,”
J. Polym. Sci. Polym. Phys. Ed.
17
(
7
),
1097
1162
(
1979
).
21.
Moynihan
,
C. T.
,
A. J.
Easteal
,
M. A.
De Bolt
, and
J.
Tucker
, “
Dependence of the fictive temperature of glass on cooling rate
,”
J. Am. Ceram. Soc.
59
(
1–2
),
12
16
(
1976
).
22.
Caruthers
,
J. M.
,
D. B.
Adolf
,
R. S.
Chambers
, and
P.
Shrikhande
, “
A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymers
,”
Polymer
45
(
13
),
4577
4597
(
2004
).
23.
Adolf
,
D. B.
,
R. S.
Chambers
, and
M. A.
Neidigk
, “
A simplified potential energy clock model for glassy polymers
,”
Polymer
50
(
17
),
4257
4269
(
2009
).
24.
Medvedev
,
G. A.
, and
J. M.
Caruthers
, “
Development of a stochastic constitutive model for prediction of postyield softening in glassy polymers
,”
J. Rheol.
57
(
3
),
949
1002
(
2013
).
25.
Kröner
,
E.
, “
General continuum theory of dislocations and proper stresses
,”
Arch. Ration. Mech. Anal.
4
,
273
334
(
1959
).
26.
Lee
,
E. H.
, “
Elastic-plastic deformation at finite strains
,”
J. Appl. Mech.
36
(
1
),
1
6
(
1969
).
27.
Lubarda
,
V. A.
, and
E. H.
Lee
, “
A correct definition of elastic and plastic deformation and its computational significance
,”
J. Appl. Mech.
48
(
1
),
35
40
(
1981
).
28.
Negahban
,
M.
, and
A. S.
Wineman
, “
Material symmetry and the evolution of anisotropies in a simple material—II: The evolution of material symmetry
,”
Int. J. Non Linear Mech.
24
(
6
),
537
549
(
1989
).
29.
Negahban
,
M.
, and
A. S.
Wineman
, “
The evolution of anisotropies in the elastic response of an elastic-plastic material
,”
It. J. Plast.
8
(
5
),
519
542
(
1992
).
30.
Tool
,
A. Q.
, “
Relaxation of stresses in annealing glass
,”
J. Res. Natl. Bur. Stand.
34
(
2
),
199
211
(
1945
).
31.
Negahban
,
M.
, “
Modeling the thermomechanical effects of crystallization in natural rubber: II: Elementary thermodynamic properties
,”
Int. J. Solids Struct.
37
(
20
),
2791
2809
(
2000
).
32.
Walsh
,
D.
, and
P.
Zoller
,
Standard Pressure Volume Temperature Data for Polymers
(
CRC
, Switzerland,
1995
).
33.
Van Krevelen
,
D. W.
, and
K.
Te Nijenhuis
,
Properties of Polymers Their Correlation with Chemical Structure; Their Numerical Estimation and Prediction From Additive Group Contributions
(
Elsevier
, Amsterdam,
2009
).
34.
Kovarskii
,
A. L.
,
High-Pressure Chemistry and Physics of Polymers
(
CRC
, Boca Raton,
1994
).
35.
Münstedt
,
H.
, “
Influence of hydrostatic pressure on rheological properties of polymer melts—A review
,”
J. Rheol.
64
(
3
),
751
774
(
2020
).
36.
Rudolph
,
N. M.
,
A. C.
Agudelo
,
J. C.
Granada
,
H. E.
Park
, and
T. A.
Osswald
, “
WLF model for the pressure dependence of zero shear viscosity of polycarbonate
,”
Rheol. Acta
55
(
8
),
673
681
(
2016
).
37.
Delbreilh
,
L.
,
E.
Dargent
,
J.
Grenet
,
J.-M.
Saiter
,
A.
Bernès
, and
C.
Lacabanne
, “
Study of poly (bisphenol A carbonate) relaxation kinetics at the glass transition temperature
,”
Eur. Polym. J.
43
(
1
),
249
254
(
2007
).
38.
Greiner
,
R.
, and
F.
Schwarzl
, “
Thermal contraction and volume relaxation of amorphous polymers
,”
Rheol. Acta
23
(
4
),
378
395
(
1984
).
39.
Delbreilh
,
L.
,
A.
Bernès
, and
C.
Lacabanne
, “
Secondary retardation/relaxation processes in bisphenol A polycarbonate: Thermostimulated creep and dynamic mechanical analysis combined investigations
,”
Int. J. Polym. Anal. Charact.
10
(
1–2
),
41
56
(
2005
).
40.
Qian
,
Z.
, and
G. B.
McKenna
, “
Expanding the application of the van Gurp–Palmen plot: New insights into polymer melt rheology
,”
Polymer
155
,
208
217
(
2018
).
41.
Delbreilh
,
L.
,
Etude de l'hétérogénéité et de la métastabilité de phases amorphes polymères par analyse des transitions et des phénomènes relaxationnels lents
2004
, Vol. 163, p.
1
.
42.
Koh
,
Y. P.
, and
S. L.
Simon
, “
The glass transition and enthalpy recovery of a single polystyrene ultrathin film using flash DSC
,”
J. Chem. Phys.
146
(
20
),
203329
(
2017
).
43.
Schultheisz
,
C. R.
, and
G. B.
McKenna
,
Volume recovery and physical aging in glassy polycarbonate following temperature jumps
, in
Proceedings North American Thermal Analysis Society, 25th Annual Meeting
, McLean, Virginia (The North American Thermal Analysis,
1997
), 366–373.
44.
Strabala
,
K. W.
,
The effects of combined compression and aging on the properties of glassy polycarbonate
, in
Engineering Mechanics
(
University of Nebraska-Lincoln
,
Lincoln, NE
,
2009
), p.
98
.
45.
Shamim
,
N.
,
Y. P.
Koh
,
S. L.
Simon
, and
G. B.
McKenna
, “
Glass transition temperature of thin polycarbonate films measured by flash differential scanning calorimetry
,”
J. Polym. Sci. Part B: Polym. Phys.
52
(
22
),
1462
1468
(
2014
).
46.
McKenna
,
G. B.
,
M. G.
Vangel
,
A. L.
Rukhin
,
S. D.
Leigh
,
B.
Lotz
, and
C.
Straupe
, “
The τ-effective paradox revisited: An extended analysis of Kovacs’ volume recovery data on poly(vinyl acetate)
,”
Polymer
40
(
18
),
5183
5205
(
1999
).
47.
Rudolph
,
N. M.
,
T. A.
Osswald
, and
G. W.
Ehrenstein
, “
Influence of pressure on volume, temperature and crystallization of thermoplastics during polymer processing
,”
Int. Polym. Process.
26
(
3
),
239
248
(
2011
).
48.
Tribone
,
J. J.
,
J. M.
O'reilly
, and
J.
Greener
, “
Pressure-jump volume-relaxation studies of polystyrene in the glass transition region
,”
J. Polym. Sci. Part B: Polym. Phys.
27
(
4
),
837
857
(
1989
).
49.
Zhao
,
X.
,
L.
Grassia
, and
S. L.
Simon
, “
Mobility of pressure-densified and pressure-expanded polystyrene glasses: Dilatometry and a test of KAHR model
,”
Macromolecules
54
(
18
),
8352
8364
(
2021
).
50.
Dreistadt
,
C.
,
A.-S.
Bonnet
,
P.
Chevrier
, and
P.
Lipinski
, “
Experimental study of the polycarbonate behaviour during complex loadings and comparison with the Boyce, Parks and Argon model predictions
,”
Mater. Des.
30
(
8
),
3126
3140
(
2009
).
You do not currently have access to this content.