viscoelasticity.ppt

Viscoelastic
Characterization
Dr. Muanmai Apintanapong

Elastic deformation - Flow behavior



 



E
.

Introduction to Viscoelasticity
All viscous liquids deform
continuously under the influence of an
applied stress – They exhibit viscous
behavior.
Solids deform under an applied stress,
but soon reach a position of
equilibrium, in which further
deformation ceases. If the stress is
removed they recover their original
shape – They exhibit elastic behavior.
Viscoelastic fluids can exhibit both
viscosity and elasticity, depending on
the conditions.
Viscous fluid
Viscoelastic fluid
Elastic solid

Viscosity =resistance to flow







Viscosity of fluids at 20C
Go to stress relaxation

Viscosity: temperature
dependence

Flow curve and Viscosity curve

Flow behavior: viscosity curve

Stress Relaxation
Universal Testing
Machine
Instron,
TA XT2
Force sensor

Stress Relaxation Test
Time, t
Strain
Stress
Elastic
Viscoelastic
Viscous fluid
0
Stress
Stress
Viscous fluid
Viscous fluid

Strain is applied to sample instantaneously
(in principle) and held constant with time.
Stress is monitored as a function of time (t).
Stress Relaxation Experiment
Strain
0
time

Stress Relaxation Experiment
Stress decreases with time
starting at some high value
and decreasing to zero.
Response of Material
time
0
Response of Classical Extremes
time
0
stress for t>0
is constant
time
0
stress for t>0
is 0
Hookean Solid Newtonian Fluid

Stress is applied to sample instantaneously, t1, and held
constant for a specific period of time. The strain is monitored
as a function of time ((t) or (t)).
The stress is reduced to zero, t2, and the strain is monitored
as a function of timetort
Creep Recovery Experiment
Stress
time
t1 t2

Response of Classical Extremes
– Stain for t>t1 is constant
– Strain for t >t2 is 0
time
time time
– Stain rate for t>t1 is constant
– Strain for t>t1 increase with time
– Strain rate for t >t2 is 0
t2
t1
t1 t2 t2
t1

Reference: Mark, J., et.al., Physical Properties of Polymers ,American Chemical Society, 1984, p. 102.
Creep Recovery Experiment:
Response of Viscoelastic Material
Creep > 0
time
t 1
t2
Recoverable
Strain
Recovery  = 0 (after steady state)
/
Strain rate decreases with
time in the creep zone,
until finally reaching a
steady state.
In the recovery zone, the viscoelastic fluid
recoils, eventually reaching a equilibrium
at some small total strain relative to the
strain at unloading.

time
Recovery Zone
Creep Zone
Less Elastic
More Elastic
Creep  >0 Recovery  = 0 (after steady state)
/
t1 t2

Rheological Models
• Mechanical components or
elements

Elastic (Solid-like) Response
A material is perfectly elastic, if the equilibrium
shape is attained instantaneously when a stress
is applied. Upon imposing a step input in strain,
the stresses do not relax.
The simplest elastic solid model is the Hookean
model, which we can represent by the “spring”
mechanical analog.

 E


Elasticity deals with mechanical properties
of elastic solids (Hooke’s Law)
 
Stress, 
L
Strain,  = L/L
L
E=/

Strain, 
Stress,





E
E
slope



 /

Elastic (Solid-like) Response
• Stress Relaxation experiment
 strain)
time
 stress)
time
to=0
• Creep Experiment
 stress)
time
to=0
 strain)
time
to=0
to=0
ts
ts
o
o/E
o

Viscous (Liquid-like) Response
A material is purely viscous (or inelastic) if following any flow or
deformation history, the stresses in the material become
instantaneously zero, as soon as the flow is stopped; or the
deformation rate becomes instantaneously zero when the stresses
are set equal to zero. Upon imposing a step input in strain, the
stresses relax as soon as the strain is constant.
The liquid behavior can be simply represented by the Newtonian
model. We can represent the Newtonian behavior by using a
“dashpot” mechanical analog:


 


Theory of
Hydrodynamics
In Newtonian Fluids, Stress is proportional to rate of
strain but independent of strain itself
Newton’s Law










Strain, , 
Stress,

dt
d
dt
d
slope
























/
/
/

Viscous (Liquid-like) Response
• Stress Relaxation experiment (suddenly applying a strain to the sample
and following the stress as a function of time as the strain is held
constant).
 strain)
time
 stress)
time
to=0
• Creep Experiment (a constant stress is instantaneously applied to the
material and the resulting strain is followed as a function of time)
tstress)
time
to=0
 strain)
time
to=0
to=0
ts
ts



 o
dt
d
slope 



s
o
t


 
to
o
0




 const
o

Energy Storage/Dissipation
• Elastic materials store energy (capacitance)
• Viscous materials dissipate energy (resistance)
t
Energy
Energy
t
E
Viscoelastic materials store and
dissipate a part of the energy
t

What causes viscoelastic behavior?
Long polymer chains at the molecular scale, make
polymeric matrix viscoelastic at the microscale
Reference: Dynamics of Polymeric Liquids (1977). Bird,
Armstrong and Hassager. John Wiley and Sons. pp: 63.
Energy
Storage
+Dissipation

• Specifically, viscoelasticity is a molecular
rearrangement. When a stress is applied to a
viscoelastic material such as a polymer, parts of
the long polymer chain change position. This
movement or rearrangement is called Creep.
Polymers remain a solid material even when these
parts of their chains are rearranging in order to
accompany the stress, and as this occurs, it
creates a back stress in the material. When the
back stress is the same magnitude as the applied
stress, the material no longer creeps. When the
original stress is taken away, the accumulated
back stresses will cause the polymer to return to
its original form. The material creeps, which gives
the prefix visco-, and the material fully recovers,
which gives the suffix –elasticity.
http://en.wikipedia.org/wiki/Viscoelasticity

Examples of viscoelastic foods:
• Food starch, gums, gels
• Grains
• Most solid foods (fruits, vegetables, tubers)
• Cheese
• Pasta, cookies, breakfast cereals
Almost all solid foods and fluid foods containing
long chain biopolymers

Viscoelasticity Experiments
• Static Tests
– Stress Relaxation test
– Creep test
• Dynamic Tests
– Controlled strain
– Controlled stress
(When we apply a small oscillatory
strain and measure the resulting
stress)

Why we want to fit models
to viscoelastic test data?
• To quantify the data – mathematical representation
For use with other food processing applications
- Some food drying models require viscoelastic
properties
- Design of pipelines, mixing vessels etc., using
viscoelastic fluid foods
• To obtain information at different test conditions
– Example: Extrusion
• To obtain an estimate of elastic properties and
relaxation times
– Helps to quantify glass transition

Viscoelastic Models
• Maxwell Model
• Kelvin-Voigt Model
Used for stress
relaxation tests
Used for creep tests

Viscoelastic Response – Maxwell Element
A viscoelastic material (liquid or solid) will not respond instantaneously
when stresses are applied, or the stresses will not respond
instantaneously to any imposed deformation. Upon imposing a step
input in strain the viscoelastic liquid or solid will show stress relaxation
over a significant time.
At least two components are needed, one to characterize elastic and the
other viscous behavior. One such model is the Maxwell model:

 E


 



Viscoelastic Response
Let’s try to deform the Maxwell
element E
Strain,
Stress,

Maxwell Model Response
• The Maxwell model can describe successfully the
phenomena of elastic strain, creep recovery,
permanent set and stress relaxation observed with
real materials
• Moreover the model exhibits relaxation of stresses
after a step strain deformation and continuous
deformation as long as the stress is maintained. These
are characteristics of liquid-like behaviour
• Therefore the Maxwell element represents a
VISCOELASTIC FLUID.

Maxwell Model-when  is applied
( )
d
Dot represents
dt
 Stress 








d
d
s
s
d
s
dt
d
E 








1.  will be same in each element











































dt
d
E
dt
d
E
and
E
E
t
t
or
s
s
d
s
s
s
d
s
d
s
1
/

2. Total  = sum of individual 

1) Creep Experiment: If a sudden stress is imposed (step loading), an
instantaneous stretching of the spring will occur, followed by an extension of
the dashpot. Deformation after removal of the stress is known as creep
recovery:
.
t
E
t o
o



 

)
(
 stress)
time
to=0

time
to=0
ts ts
o
o/E

o
slope 
s
o
t


Or by defining the “creep compliance”:
o
t
t
J

 )
(
)
( 

t
E
t
J 

1
)
(
Elastic Recovery
Permanent
Set
o/E
dashpot
spring

2) Stress Relaxation Experiment: If the mechanical model is suddenly
extended to a position and held there (o=const., =0):
t

 /
)
( t
oe
t 
 Exponential decay
strain)
time
to=0
.
o
t
/
t
e
 o
G
G(t)
Also recall the definition of the “relaxation” modulus:
o
t
t
G

 )
(
)
( 
  t

 /
)
( t
o
o e
G
t 
 and
 stress)
time
to=0
o=Goo   t

 /
)
( t
o e
t 

t = /E = Relaxation time = the time required by biopolymers to relax the stresses

Generalized Maxwell Model
The Maxwell model is qualitatively reasonable, but does
not fit real data very well.
Instead, we can use the generalized Maxwell model
t1 t 2 t 3
t n
E1 E2 E3 En

n
Applied for stress relaxation test
)
n
n
t
n
t
t
t
n
t
t
e
E
e
E
e
E
t
e
e
e
t
t
t
t
t
t
t






/
/
2
/
1
0
/
/
2
/
1
.....
(
)
(
.....
)
(
2
1
2
1













Determination of parameters for
• There are 4 methods.
– Method of Instantaneous Slope
– Method of Central Limit Theorem
– Point of Inflection Method
– Method of Successive Residuals
 direct method and more popular
Optional

Method of Successive Residuals
• First plot-semilog plot: if it is linear,
use single Maxwell Model
• If it is not linear, use Generalized
Maxwell Model

Divided into many parts and plot of each part until the
curvature disappears.
1
1
1
1
/
1
/
ln
)
(
ln
)
( 1
t

t



 t
1/
-
slope
ln
intercept
-
y




 
t
t
e
t t
time
to=0
ln  Second plot
Slope of straight line = 1/t2
Plot until it is straight
2
2
2
2
/
2
/
ln
)
(
ln
)
( 2
t

t



 t
1/
-
slope
ln
intercept
-
y




 
t
t
e
t t
time
to=0
ln 
First plot
Slope of straight line = -1/t1
ln 1
ln 2

Example: Genealized maxwell model
for stress relaxation test
• Test sample has 2 cm
diameter and 4 cm long
Area = 3.142 X 10-4 m2
t (min) F (kg)
0 100
0.5 74
1 66.5
1.5 61
2 57
2.5 54.5
3 53
3.5 51.5
4 51
4.5 50
5 49
6 48.5
7 47.5
8 47
9 46
10 45
11 44
12 43

10000
100000
1000000
10000000
0 100 200 300 400 500 600 700 800
time (s ec)
s
tres
s
(Pa)
first plot
ln 1= 14.344238 =y-intercept
1= 1696771.4
slope= -0.000318
t1= 3143.6655
second plot
ln 2= 14.183214 =y-intercept
2= 1444413.62
slope= -0.0198653
t2= 50.3390334
Model
Maxwell
Two
e
e
t t
t 34
.
50
/
6
67
.
3143
/
6
10
444
.
1
10
697
.
1
)
( 







first plot second plot
t(s) stress Pa stress Pa
0 3121364 1424592.282
30 2309809 629153.0092
60 2075707 411012.9526
90 1904032 255148.5658
120 1779177 145954.4705
150 1701143 83432.09296
180 1654323 51976.02755
210 1607502
240 1591895
270 1560682
300 1529468
360 1513861
420 1482648
480 1467041
540 1435827
600 1404614
660 1373400
720 1342186

Voigt-Kelvin Model Response
• The Voigt-Kelvin element does not continue to deform as long as
stress is applied, rather it reaches an equilibrium deformation. It does
not exhibit any permanent set. These resemble the response of cross-
linked rubbers and are characteristics of solid-like behaviour
• Therefore the Voigt-Kelvin element represents a VISCOELASTIC
SOLID.
 The Voigt-Kelvin element cannot describe stress relaxation.
 Both Maxwell and Voigt-Kelvin elements can provide only a qualitative
description of the response
 Various other spring/dashpot combinations have been proposed.

Viscoelastic Reponse
Voigt-Kelvin Element
The Voigt-Kelvin element consists of a
spring and a dashpot connected in parallel.

















E
dashpot
spring
dashpot
spring
E


Creep Recovery Experiment:
applied 0 (step loading)

strain)
time
to=0
o
+

strain)
time
to=0
t = /E = characteristic time = time of retardation
t

 /
0
)
( t
e
E
t 

time
t 0
t
Slope=
/
0/E
 
t

 /
0
1
)
( t
e
E
t 



Generalized Voigt-Kelvin Model
E1
E2
E3
En
i
i
i
E

t 
 
i
t
n
i i
e
E
t t

 /
1
0 1
1
)
( 


 

Three element Model
• Standard linear solid

Four element Model
E1
E2 2
1

strain)
time
to=0
o

strain)
time
to=0

strain)
time
to=0
spring
Kevin
dashpot
 
1
0
/
2
0
1
0
1
)
(




 t t
e
E
E
t t



 

C B
A
.
Creep test: use 4-element model
 
2
2
1
0
/
2
0
1
0
1
)
(
E
t
e
E
E
t
ret
t

t




 t




 
dt
t
d
slopeAC
a
a
E
BC
AB
)
(
tan 1
0
2
0










time
t0
Strain (t)
0/E2 =r
0/E1= 0

a = 2/E2=tret
.
. Dashpot, 1
Kelvin, 2/E2
Spring, E1
Slope = 0/1

 
 
t

t





t











t
t
1
0
2
0
1
0
2
0
1
0
/
2
0
1
0
1
0
/
2
0
1
0
)
(
1
0
1
)
(
:
1
)
(
:














































E
dt
t
d
E
t
dt
d
e
E
dt
d
E
dt
d
dt
t
d
diff
t
e
E
E
t
from
ret
t
t
ret
ret

Generalized four-element model
• Combination of four-element
model in series

Example
• Analyze the given experimental creep curve
in terms of the parameters of a 4-element
model.
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7 8 9 10
time (s)
deformation
(cm)
•Cylindrical specimen
(2 cm in diameter and
5 cm long)
•Applied step load is
10 kg.

y = 0.0266x + 0.325
R2
= 0.9993
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7 8 9 10
time (s)
deformation
(cm
0/E2
r=
0.125
a 2/E2tret=0.9
0/E10
=0.2
slope =
0/1=0.0266
dashpot
kelvin
spring

Length = 0.05 m
Diameter = 0.02 m
Area = 0.00031429 m2
Load = 10 kg
0 = 312136.364 Pa
slope = Deformation/time = 0.0266 cm/s
slope = 0.00532 per sec
0/1= 0.00532 per sec
1= 58672248.8 Pa s
0/E2= 0.125 cm = 0.025 m/m
E2 = 12485454.5 Pa
a = 0.9 = 2/E2
2= 11236909.1 Pa s
0= 0/E1 = 0.2 cm = 0.04 m/m
E1 = 7803409.09 Pa

viscoelasticity.ppt

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