Bob Cohen Celebration Symposium
MIT
May 14, 2016
Viscoelasticity and Other Remembrances
Jim Caruthers
(Eldest Son of the Bob Cohen Academic Family)
Purdue University
Bob’s and My First MIT Paper
Version 2 of Creep Rig
Purdue University
Temperature
T-Jump Experiment
Time
Activation Energy
Creep
response
Still produces some of the
best nonlinear creep data for
polymeric rubbers and
glasses in the world
Nonisothermal Rheology
Nonisothermal Linear Creep Response of a Viscoelastic Material
Nonlinear Viscoelastic Behavior
Temperature Clock
dζ
t *− u * =
∫
u aT (ζ )
t
aT(t) includes effect of
temperature history
Material Clock
Temperature
Volume
Stress
Strain
Entropy
Internal Energy
Etc.
dζ
t *− u * =
∫
u a (ζ )
t
a(t) includes effect of temperature
and deformation history
Some Relaxation Experiments on
Polymeric Glasses
(Kovacs, 1963)
Nonlinear stress strain with post yield
softening/hardening (Haward, ed. 1973)
V
Volume relaxation
•
•
•
•
T
•
•
•
•
CpN
3.5
2.5
1.5
0.5
-0.5
360
370
380
Temperature (K)
Linear Viscoelasticity
temperature
heating rate
cooling rate
sub-Tg
annealing time
390
engineering stress (MPa)
4.5
1hr
4hr
16hr
66hr
Stress relaxation change in ordering
with strain rate
(Kim, Medvedev and
Caruthers, 2013)
ε
(Ferry, 1980)
1.00
0.90
strain
0.80
0.75
30
60
12
8
4
0
100
1.0
(a)
0
(a)
16
slow
0.95
0.85
20
0
(σ−σinf) / (σ0−σinf)
5.5
Hodge, 1982)
(σ−σinf) / (σ0−σinf)
Enthalpy relaxation
strain rate
temperature
annealing
extension vs.
compression
90
time (s)
120
150
180
200
300
ε
time (s)
400
500
600
fast
(b)
0.9
strain
0.8
0.7
0.6
0.5
0.4
0
50
100
150
time (s)
200
250
300
Nonlinear Viscoelastic Constitutive
Model Predictions
828/DEA Epoxy Resin
Adolf, Chambers & Caruthers, Polymer, 2004
Volume Relaxation
Stress-Strain
0
-0.015
-0.02
60
40
20
40
50
60
70
80
0
0.02
0
-0.02
-0.04
-0.06
50
100
time (s)
0.05
0.1
up
dow n
Model
1.4
0
0.15
20
150
80
100
120
2.6
10 7
10
60
Enthalpy-CoolingAging-Reheating
10 8
6
40
temperature (C)
10 9
shear relaxation modulus (Pa)
36 C
50 C
63 C
0.04
1.6
Physical Aging
Strain Reversal - Torsion
0.1
0.06
1.8
engineering strain
temperature (C)
0.08
5 C/min
No Aging
2
1.2
0
90
heat capacity (J/g/C)
30
2.2
heat capacity (J/g/C)
-0.01
-0.025
torque (N-m)
comp data
comp data
comp data
tens data
80
stress (MPa)
volume strain
-0.005
0
63C
50C
36C
63C
100
Enthalpy-CoolingReheating
68.6C Master Curve
Equilibrated at 73C
After 1 min at 63C
After 1 hr at 63C
After 6 hrs at 63C
10 -4 10 -3 10 -2 10 -1
10 0
10 1
time (s)
10 2
10 3
10 4
2.4
up
2.2
dow n
Model
2
1.8
1.6
5 C/min
2 hrs at 58C
1.4
1.2
0
20
40
60
80
temperature (C)
100
120
• Material clock constitutive models for glassy polymers can describe some
phenomena, but have difficulty with
1. post-yield softening and its dependence upon annealing time
2. Kovacs τeff-paradox in specific volume relaxation
3. Stress-memory experiment
4. Loading-unloading deformation
5. Tertiary creep
Something seems to be missing
• Two implicit assumptions in all traditional material clock constitutive models
1. Time shift invariance – i.e. the location of the relaxation spectra can
move with temperature and/or deformation, but the shape is invariant
2. The continuum postulate – a spatial and temporial average of all
quantities is performed PRIOR to specification of a constitutive
relationship
Simple Stochastic Constitutive Model
A model of the Glass must at least acknowledge and preferably
predict the evolution of dynamic heterogeneity
1D Volume Stochastic Model (VSM); Medvedev and Caruthers Macromol. (2012) 45, 7237
Stochastic: single relaxation time volume relaxation
 C2

v − ve
dT
log aˆ C1 
− 1
dt + ∆α
dt + N ( v, T ) dW =


aˆ ( v, T )τ 0
dt
 C2 + Θ 

e fluctuating
deterministic
dvˆ = −
Θ ( vˆ, T ) = T − Tg +
v−v
dT
dv = −
dt + ∆α
dt + N ( v, T ) dW
Weiner
noise)

a ( vProcess
, T )τ(white
dtLocal, not global
v = vˆ
0

fluctuating
deterministic
vˆ − ve (T )
v gα l − s
Stochastic (SCM); Medvedev and Caruthers J Rheol (2013)
949-1002
New 57,
term
that acknowledges
dt 
∂
1

ˆ
ln
d ( σˆ − σ ∞ ( ε ) ) =
−
σ
+
a
+
(
)

aτ 0 
∂σˆ
aτ 0
σ = σˆ
ˆ
2 dW
+ I  ∆Ktr ( ε ) dt + ∆Adθ  + G : ε d dt
dynamic heterogeneity
e

log aˆ C1  c 0 − 1
=
 eˆc

Local, not global
(
eˆc =ec∞ + θ Sˆ − S ∞
+
)
V0
σˆ d − σ ∞d ) : ( σˆ d − σ ∞d )
(
4∆G
Key nonlinearity in the SCM is the local mobility depends upon the local state of
the system, i.e. the average stress, temperature, volume, entropy, energy, etc.)
Post-yield Softening
Stress-strain curves exhibit significant post-yield softening
• Increases with physical aging
• Similar behavior in uniaxial extension and compression
4
2
20
1
O hrs
10
0
0.05
0.10
strain
0.15
0.20
Strain
0.0%
2.0%
4.0%
yield - 6.6%
8.6%
19.7%
0.4
20
0.2
O
hr
10
0
A
equilibrium
3 hr
30
5 days
pdf
3
A
stress (MPa)
stress (MPa)
A
10 days
30
Why does post-yield
softening occur??
Predictions of SCM3
PMMA – compression
0
0
0.05
0.10
strain
0.15
0.20
0
-5
0
loga
5
T = Tg - 20°C
ε = 1.6x10-4 s-1
1. Y.-W. Lee, PhD Thesis, Purdue University, 2011
2. D.B. Adolf, et al. Polymer, 45, 4599, (2004).
3. G.A. Medvedev, J.M. Caruthers, J. Rheol., 57, 949, (2013).
Shape of the distribution
changes during deformation
Constitutive (tensorial) Models for Glassy Polymers
ε2
σ3
NLVE
t* =∫
Uc4
Viscoplastic
σ5
Stochastic
local6



?

?
†
?    ?

 

Etc.
Creep Recovery
Tertiary Creep
Etc.
Stress memory
2-Step Strain
Rate
3-Step
Reloading
Unloading
ε rate reversal
of nonlinear σ(t)
Post-yield
Hardening
Post-yield
Softening
? ?


Work in
Progress
t*=f( )



 
?
Stress
Control
Strain Control
σ(ε), σY

NLVE
Linear viscoelastic
V1
Enthalpy relaxation
Model and
variable
controlling
mobility
Volume relaxation
Non-linear mechanical

 
† New phenomenological function added for post-yield relaxation
1 Knauss and Emri (1987)
2 Popelar and Liechti (1997)
3 Schapery (1969)
4 Thermo-Visco-Elastic, Caruthers et al. (2004)
5 Argon, Park, and Boyce (1988), Boyce et al (1995);
Anand and Gurtin, Anand et al. (2003); Buckley et al.
(1996); Govaert, Tervoort et al. (1995)
6 Medvedev and Caruthers (2013)
Stochastic: single relaxation time volume relaxation
v − ve
dT
dv = −
dt + ∆α
dt + N ( v, T ) dW

a ( v, T ) τ 0
dt

fluctuating
deterministic
New term that acknowledges
dynamic heterogeneity
• Very simple model (in 3D tensorial form)
• Provides the best description of the nonlinear viscoelastic behavior
of glassy polymers to date.
• Key physics: local rate of relaxation is given by the local mobility
What I learned from Bob Cohen
Bob Cohen
Subsequent Extensions
Nonlinear Creep Experiments
Basic Continuum Mechanics
Fluctuations
Linear Viscoelasticity
Finite Viscoelasticity
Time-Temperature Superposition
Thermorheological Complexity
Nonisothermal Material Clock
Deformation Material Clock
The excitement of exploring the
behavior of polymers
Remembrances of Bob in the yearly days at MIT
1973 - 1977
Bob Cohen
Great Teacher
Incredible Mentor
Friend
Thank you !!!!!