Chris Macosko
Department of Chemical Engineering and Materials Science
NSF

MRSEC
(National Science Foundation sponsored Materials Research Science and Engineering Center)
IP
RIME
(Industrial Partnership for Research in Interfacial and Materials Engineering)
IMA Annual Program Year Tutorial
An Introduction to Funny (Complex) Fluids: Rheology, Modeling and Theorems
September 12

13, 2009
Understanding silly putty, snail slime
and other funny fluids
What is rheology?
rein
(Greek)
=
ta panta rei
=
rheology =
honey and
mayonnaise
rate of deformation
stress=
f/area
to flow
every thing flows
study of flow?, i.e. fluid mechanics?
honey and
mayo
rate of deformation
viscosity =
stress/rate
What is rheology?
rein
(Greek)
=
ta panta rei
=
rheology =
rubber band and
silly putty
time of deformation
modulus
= f/area
to flow
every thing flows
study of flow?, i.e. fluid mechanics?
honey and
mayo
rate of deformation
viscosity
4 key rheological phenomena
fluid mechanician:
simple fluids complex flows
rheologist:
complex fluids simple flows
materials chemist:
complex fluids complex flows
rheology =
study of deformation of complex materials
rheologist fits data to constitutive equations which

can be solved by fluid mechanician for complex flows

have a microstructural basis
from: Rheology: Principles, Measurement and Applications, VCH/Wiley (1994).
ad majorem Dei gloriam
Goal: Understand Principles of Rheology:
(stress, strain, constitutive equations)
stress = f (deformation, time)
Simplest constitutive relations:
Newton’s Law:
Hooke’s Law:
•
shear thinning (thickening)
•
time dependent modulus G(t)
•
normal stresses in shear N
1
•
extensional > shear stress
u
>
Key Rheological Phenomena
1

8
ELASTIC SOLID
1
The power of any spring
is in the same proportion
with the tension thereof.
Robert Hooke (1678
)
f
L
L
´
k
1
k
2
L
´
k
1
k
2
f
f
k
L
modulus
stress
t
Young (1805)
strain
1

9
Uniaxial Extension
Natural rubber
G=3.9x10
5
Pa
a = area
natural rubber G = 400 kPa
1

10
Silicone rubber G = 160 kPa
Goal: explain different results in extension and shear
obtain from Hooke’s Law in 3D
If use stress and deformation tensors
= 0
=

0.4
= 0.4
Shear gives different stress response
1

11
Stress Tensor

Notation
direction of stress on plane
plane stress acts on
Other notation besides
T
ij
:
s
ij
or
P
ij
dyad
1

12
1. Uniaxial Extension
Rheologists use very simple
T
T
22
=
T
33
= 0
or
T
22
=
T
33
T
22
T
33
T
11

T
22
causes deformation
T
11
=
0
1

13
Consider only normal stress components
Hydrostatic Pressure
T
11
= T
22
= T
33
=

p
If a liquid is
incompressible
G ≠ f(p)
≠ f(p)
Then only
t
the
extra
or
viscous
stresses
cause deformation
T
=

p
I
+
t
and only the normal stress
differences
cause deformation
T
11

T
22
=
t
11

t
22
≡ N
1
(shear)
1

14
2. Simple Shear
But to balance angular momentum
Stress tensor for simple shear
Only 3 components:
T
12
T
11
–
T
22
=
t
11
–
t
22
≡ N
1
T
22
–
T
33
=
t
22
–
t
33
≡ N
2
in general
T
21
T
21
T
12
T
12
Rheologists use very simple
T
Hooke
→
Young
→
Cauchy
→
Gibbs
Einstein
(1678)
(~1801)
(1830’s)
(1880,~1905)
Stress Tensor Summary
T
11
T
11
n
2. in general
T
= f( time or rate, strain)
3. simple
T
for rheologically complex materials:

extension and shear
4.
T
= pressure + extra stress
=

p
I
+
t
.
5.
τ
causes deformation
6. normal stress differences cause deformation,
t
11

t
22
=
T
11

T
22
7. symmetric
T
=
T
T
i.e.
T
12
=
T
21
1.
stress at point
on any
plane
1

16
Deformation Gradient Tensor
a new tensor !
s = w
–
y
w = y + s
w
y
s
P
Q
s′ = w′

y′
w’ = y’ + s’
w′
y′
s′
P
Q
x =
displacement function
describes how material points move
s’
is a vector connecting
two very close points in the
material, P and Q
1

17
Apply
F
to Uniaxial Extension
Displacement functions describe how coordinates of P in
undeformed state, x
i
‘
have been displaced to coordinates of P in
deformed state, x
i
.
1

18
Can we write Hooke’s Law as ?
Assume:
1)
constant volume V′ = V
2) symmetric about the x
1
axis
1

19
Can we write Hooke’s Law as ?
Solid Body Rotation
–
expect no stresses
For solid body rotation,
expect
F
=
I
t
=0
䉵B
F
≠
I
F
≠
F
T
Need to get rid of rotation
create a new tensor!
1

20
B
ij
gives relative local change in area within the sample.
Finger Tensor
Solid Body Rotation
1

21
1. Uniaxial Extension
since
T
22
= 0
Neo

Hookean Solid
1

22
2. Simple Shear
agrees with experiment
Silicone rubber G = 160 kPa
1.
area change around a point
on any
plane
2. symmetric
3. eliminates rotation
4. gives Hooke’s Law in 3D
fits rubber data fairly well
predicts N
1
, shear normal stresses
Finger Deformation Tensor Summary
Course Goal: Understand Principles of Rheology:
(constitutive equations)
stress = f (deformation, time)
Simplest constitutive relations:
Newton’s Law:
Hooke’s Law:
•
shear thinning (thickening)
•
time dependent modulus G(t)
•
normal stresses in shear N
1
•
extensional > shear stress
u
>
Key Rheological Phenomena
VISCOUS LIQUID
2
The resistance which arises
From the lack of slipperiness
Originating in a fluid, other
Things being equal, is
Proportional to the velocity
by which the parts of the
fluids are being separated
from each other.
Isaac S. Newton (1687)
measured
in shear
1856
capillary (Poiseuille)
1880’s concentric cylinders
(Perry, Mallock,
Couette, Schwedoff)
Newton, 1687
Stokes

Navier, 1845
Bernoulli
Familiar materials have a wide range in viscosity
Adapted from
Barnes et al.
(1989).
measured
in shear
1856
capillary (Poiseuille)
1880’s concentric cylinders
(Perry, Mallock,
Couette, Schwedoff)
measured in extension
1906
Trouton
u
= 3
Newton, 1687
Stokes

Navier, 1845
Bernoulli
To hold his viscous pitch samples, Trouton forced a
thickened end into a small metal box. A hook was
attached to the box from which weights were hung.
“A variety of pitch which gave by the traction method
l
= 4.3 x 10
10
(poise) was found by the torsion
method to have a viscosity
m
= 1.4 x 10
10
(poise).”
F.T. Trouton (1906)
polystyrene 160
°
C
Münstedt (1980)
Goal
1.
Put Newton’s Law in 3 dimensions
•
rate of strain tensor 2
D
•
show
u
= 3
Separation and displacement
of point Q from P
s = w

y
s′ = w′

y′
w′
y′
s′
P
Q
recall Deformation Gradient Tensor, F
w
y
s
P
Q
Alternate notation
:
Velocity Gradient Tensor
Viscosity is “proportional to the
velocity
by which the parts of the
fluids are being
separated
from each other.”
—
Newton
Can we write Newton’s Law for viscosity as
t
=
L
?
solid body rotation
Rate of Deformation Tensor
D
t
2
≠
t
2
Other notation:
Vorticity Tensor
W
Example 2.2.4 Rate of Deformation Tensor is a Time Derivative of
B
.
Show that 2D = 0 for solid body rotation
Here planes of fluid slide over each other like cards in a deck.
Steady simple shear
Newtonian Liquid
t
=
2
D
or
T
=

p
I
+
2
D
Time derivatives of the displacement functions
for simple, shear
Steady Uniaxial Extension
Newtonian Liquid
Apply to Uniaxial Extension
t
=
2D
From definition of extensional viscosity
Newton’s Law in 3 Dimensions
•
predicts
0
low shear rate
•
predicts
u0
= 3
0
but many materials show large deviation
Newtonian Liquid
T
11
T
11
n
1.
stress
at point on
plane
Summary of Fundamentals
simple
T

extension and shear
T
= pressure + extra stress
=

p
I
+
t
.
symmetric
T
=
T
T
i.e.
T
12
=
T
21
2.
area change
around a point on plane
symmetric, eliminates rotation
gives Hooke’s Law in 3D, E=3G
3.
rate of separation
of particles
symmetric, eliminates rotation
gives Newton’s Law in 3D,
Course Goal: Understand Principles of Rheology:
stress = f (deformation, time)
NeoHookean: Newtonian:
•
shear thinning (thickening)
•
time dependent modulus G(t)
•
normal stresses in shear N
1
•
extensional > shear stress
u
>
Key Rheological Phenomena
t
=
2
D
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