Viscoelastic Characterization

Dr. Muanmai Apintanapong

Elastic deformation - Flow behavior

Elastic behavior

Newtonian behavior

Newtonian liquid

E.

Introduction to ViscoelasticityAll 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

Shear Stress

A

F

Shear Rate

Practical shear rate values

Viscosity =resistance to flow

Viscosity of fluids at 20C

Go to stress relaxation

Viscosity: temperature dependence

Flow curve and Viscosity curve

Flow behavior: flow 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

StressStress

Viscous fluidViscous 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

Str

ain

0 time

Stress Relaxation Experiment

Stress decreases with timestarting at some high value and decreasing to zero.

Response of Material

Str

ess

time0

Response of Classical Extremes

time0

stress for t>0is constant

time0

stress for t>0 is 0S

tres

s

Str

ess

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 timetort

Creep Recovery Experiment

Str

ess

timet1 t2

Creep Recovery Experiment

Response of Classical Extremes

– Stain for t>t1 is constant– Strain for t >t2 is 0

time

Str

ain

time

Str

ain

time

– Stain rate for t>t1 is constant– Strain for t>t1 increase with time– Strain rate for t >t2 is 0

t2S

tre

sst1

t1 t2 t2t1

Reference: Mark, J., et.al., Physical Properties of Polymers ,American Chemical Society, 1984, p. 102.

Creep Recovery Experiment:Response of Viscoelastic Material

Creep > 0

timet 1 t2

RecoverableStrain

Recovery = 0 (after steady state)

Str

ain

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 ZoneCreep Zone

Less Elastic

More Elastic

Creep 0 Recovery = 0 (after steady state)

/

Str

ain

t1 t2

Creep Recovery Experiment

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,

Str

ess,

E

Eslope

/

Elastic (Solid-like) Response• Stress Relaxation experiment

strain)

time

stress)

timeto=0

• Creep Experiment

stress)

timeto=0

strain)

timeto=0

to=0

tsts

oo/E

o

Viscous (Liquid-like) ResponseA 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, ,

Str

ess,

dt

d

dt

dslope

///

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)

timeto=0

• Creep Experiment (a constant stress is instantaneously applied to the material and the resulting strain is followed as a function of time)

stress)

timeto=0

strain)

timeto=0

to=0

tsts

o

dt

dslope

so t o

o

0

consto

Energy Storage/Dissipation• Elastic materials store energy (capacitance)

• Viscous materials dissipate energy (resistance) t

Energy

Energy

t

E Viscoelastic materials store Viscoelastic materials store

andand dissipate a part of the dissipate a part of the energyenergy

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 str

ain and measure the resulting stress)

Why we want to fit models to viscoelastic test data?

• To quantify the data – mathematical representationFor 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 representsdt

Stress

dd

ss

ds

dt

dE

1. will be same in each element

dt

d

Edt

d

E

andEE

t

t

or

ss

d

ss

s

ds

ds

1

/

2. Total = sum of individual

Maxwell Model Response1) 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:

.

tE

t oo

)(

stress)

timeto=0

timeto=0ts ts

oo/E

oslope

so t

Or by defining the “creep compliance”:o

ttJ

)(

)(

t

EtJ

1)(

Elastic Recovery

Permanent Set

o/Edashpot

spring

Maxwell Model Response2) Stress Relaxation Experiment: If the mechanical model is suddenly

extended to a position and held there (=const., =0):

/)( toet Exponential decay

strain)

timeto=0

.

o

/te oGG(t)

Also recall the definition of the “relaxation” modulus:o

ttG

)(

)(

/)( too eGt and

stress)

timeto=0

o=Goo /)( to et

= /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

1 2 3

nE1 E2 E3 En

n

Generalized Maxwell Model

Applied for stress relaxation test

)

n

n

tn

tt

tn

tt

eEeEeEt

eeet

//

2/

10

//2

/1

.....()(

.....)(21

21

Determination of parameters for Generalized Maxwell Model

• 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

11

/1

/ln)(ln

)( 1

1/ - slope

ln intercept-y

tt

et t

timeto=0

ln Second plot

Slope of straight line = 1/2

Plot until it is straight

2

2

22

/2

/ln)(ln

)( 2

1/ - slope

ln intercept-y

tt

et t

timeto=0

ln First plot

Slope of straight line = -1/1

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 (sec)

stre

ss (

Pa)

first plot

ln 1= 14.344238 =y-intercept

1= 1696771.4

slope= -0.000318

1= 3143.6655

second plot

ln 2= 14.183214 =y-intercept

2= 1444413.62

slope= -0.0198653

2= 50.3390334

ModelMaxwellTwo

eet tt 34.50/667.3143/6 10444.1 10697.1)(

  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 ReponseVoigt-Kelvin Element

The Voigt-Kelvin element consists of a spring and a dashpot connected in parallel.

E

dashpotspring

dashpotspring

E

Creep Recovery Experiment: applied 0 (step loading)

strain)

timeto=0

o

+strain)

timeto=0

= /E = characteristic time = time of retardation

/0)( teE

t

timet 0 t

Slope=

Str

ain (

t)

0/E

/0 1)( teE

t

Generalized Voigt-Kelvin Model

E1

E2

E3

En

i

ii E

itn

i i

eE

t /

10 1

1)(

Three element Model

• Standard linear solid

Four element Model

E1

E2 2

1

strain)

timeto=0

o

strain)

timeto=0

strain)

timeto=0

spring

Kevin

dashpot

1

0/

2

0

1

0 1)( t

eEE

t t

C B

A

.

Creep test: use 4-element model

2

2

1

0/

2

0

1

0 1)(

E

te

EEt

ret

t

dt

td

slopeACa

aE

BC

AB

)(

tan 1

0

2

0

timet0

Strain (t)

0/E2 =r

0/E1= 0

a = 2/E2=ret

.. Dashpot, 1

Kelvin, 2/E2

Spring, E1

Slope = 0/1

1

0

2

0

1

0

2

0

1

0/

2

0

1

0

1

0/

2

0

1

0

)(

10

1)(

:

1)(:

E

dt

td

E

t

dt

de

Edt

d

Edt

d

dt

tddiff

te

EEtfrom

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)

r

aret

0

=0.2

slope = =0.0266

dashpot

kelvin

spring

Length = 0.05 m

Diameter = 0.02 m

Area = 0.00031429 m2

Load = 10 kg

312136.364 Pa

slope = Deformation/time = 0.0266 cm/s

slope = 0.00532 per sec

0.00532 per sec

58672248.8 Pa s

0.125 cm = 0.025 m/m

12485454.5 Pa

a = 0.9 = 11236909.1 Pa s

0/E1 = 0.2 cm = 0.04 m/m

7803409.09 Pa