Mechanical Properties Polymer_141013

7/27/2019 Mechanical Properties Polymer_141013

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Mechanical Properties ofMechanical Properties of

PolymersPolymers


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Deformation of Materialsplastic rubber

Lo

L

L

Effective spring constant of the material: LFk = /Material dependent property (modulus):

0/

/

LL

AFM

=

Youngs modulus (E) specifies resistance of the material to elongations at smalldeformations.


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Dependence of the Modulus vs

Temperature


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Dependence of the Modulus vsTime

Schematic modulus-time curve for a polymer at constant temperature


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Specific-volume vs Temperature

Specific-volume data for poly(vinyl acetate) used to determine

glass transition temperature Tg.


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Phenomenological TreatmentPhenomenological Treatmentofof ViscoelasticityViscoelasticity


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Elastic ModulusStress force per unit area

YZ

F

A

FE ==

Units: dynes/cm2 (dynes per square centimeter), or N/m2 (Newton per square meter)

FF

Y

ZX X

Tensile strain resulted form applicationof uniaxial stress

XX=

Tensile modulus and the tensile compliance:

DE E

1==


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Elastic Modulus

F

FY

Z

X

XXShear deformation

Shear stress:

ZX

F=

Shear strain:

tan=

= YX

Shear modulus G and shear complianceJare defined as

JG

1==


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Deformation at Constant Volume

Initial state

Deformed stateF F

Material stress ij is a stress produced by the deformed material.

x

xxx

AF=

Fx x

y

zAx


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Deformation at Constant VolumeForce balance at the boundary

xxAxpAx

xxAx

pxxxx +=

or rearranging

pxxxx =

pAy

yyAypyy =

yyxxxx =


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Deformation at Constant VolumeHooks law

ijij G =

where ij is a deformation tensor

The definition of the deformation should be consistent with

the shear experiment

i

j

j

iij

xu

xu

+

=

where x1

=x, x2

=y and x3

=z

2=

+

=

x

u

x

u xxxx

Tensile experiment

=

=

+

=

y

u

y

u

y

u yyyyy 2


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Deformation at Constant Volume

Relation between Shear and Youngs modulus

( ) GGG yyxxyyxxxx 3)(2 ====

xxE

E ==

GE 3=

Using the definition of the Youngs modulus

Sample does not change volume!!!


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Poissons Ratio1

1

1

1+ 1-

1-

Volume change upon deformation

)21(1)1)(1( 2 +=VNo volume change when the Poissons ratio =1/2

General relation between Shear and Youngs modulus

)1(2 +

=E

G


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Elastic Properties of Polymers as Compared to Other

Materials (T=300K)


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Understanding the mechanical

response on the molecular levelThe elastic properties of solids (glasses) is a result of the

Intermolecular forces between the atoms.

a

Deformed sample

x

The force acting between atoms at small

deformations is equal to

)( axkF =The area per atom is equal to a2 and thetensile stress is written as

Ea

ax

a

k

a

axk

=

=

=)()(

2

E=k/aYoungs Modulus


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Understanding the mechanical

response on the molecular level

The constant k is obtained by expanding intermolecular potential

around the equilibrium separation a

2

)(...

)()(

2

1)()(

2

2

22 axkconst

dx

xdUaxaUxU

ax

+=++=

=

Thus, we find

axdx

xdUk

=

=2

2)(

For a potential U(x)=f(x/a) which has a minimum value atx=a

)1()( ''

2

int

2

2

fadx

xdUk

ax

==

=

)1(''3

int fa

E

=


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Molecular Rearrangements

Liquid t~ 10-10 sec

Solids t>>100 sec

Polymer

chaint~100 sec

t~ 10-9 sec

T>>Tg


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Linear Viscoelasticity

v

The Newtonian liquid obeys the equation

dt

d

dt

d

==

where is the viscosity of liquid

Viscose Response of a Liquid

Linear viscoelasticity = elastic + viscosev

v

e

e

v

e

e

dt

dG

+=


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Readings:

Ch. 1,2 p.p.1-33, M. T. Swah, W. J. MacKnight, Introduction toPolymer Viscoelasticity.

Ch. 6. p.p. 162-174 D. I. Bower, An Introduction to Polymer Physics


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Creep response of material to a constant stress, .

Strain

Time

Solid

/Geq

Liquid.

Jeq

In these experiments the strain is monitored

as function of time, (t).

The creep complianceJ(t) is defined as

the ratio of the time dependent strain (t)

and the applied stress

)()(

ttJ

Stress-relaxation - response of materials to a constant strain, .

Stres

s

Time

Solid Geq

Liquid

In these experiments the stress is monitored

as function of time, (t).

The stress relaxation modulus G(t) is definedas the ratio of the time dependent stress (t)

and the applied strain

)(

)(

t

tG


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Mechanical Models

of Viscoelastic Materials


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The Maxwell ModelThis model consists of a spring and dashpot in series

s, s, G

d, d, For elements connected in series

For springss

G =

For the dashpotdt

d dd

=

ds == and ds +=

+=

dt

d

Gdt

d 1


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The Maxwell Model

Creep experiment: deformation at constant stress d/dt=0

=+= dtdGdtd 1

t

GtJ

ttt +==+=

1)(

)()(

0

00

Solving the equation with initial conditions at time t=0

This illustrates that the compliance of the Maxwell model increases

without limit, a behavior characteristic for viscoelastic fluids.


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The Maxwell Model

Creep experiment:

(t)

(t)

0

0

Time


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The Maxwell ModelStress relaxation experiment: deformation at constant strain d/dt=0

=+= dt

d

Gdt

d

Gdt

d 11

Solving the equation

G

dtdt

Gd=== where,

==

t

tdt

t exp)(ln))(ln( 00

The stress decays to zero at infinite time!!!

=

==

tG

tttG expexp

)()(

0

0

0


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The Maxwell ModelThe response of the Maxwell model in a stress relaxation

experiments corresponds to an elastic solid at t.

>>


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The Maxwell ModelStress-relaxation experiment:

(t)

(t)

0

G0

Time


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The Voigt ModelThis model consists of a spring and dashpot in parallel

s, s, G

d, d,

For elements connected in parallels

For springss G =

For the dashpotdt

d dd

=

ds += and ds ==

dt

dG

+=


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The Voigt ModelCreep experiment: deformation at constant stress =0

dt

dG

+=0

0 Solution of this equation is

Gwhere,exp1)( 0 =

=t

Gt is retardation time


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The Voigt ModelCreep experiment:

(t)

(t)

0/G

0

Time


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The Voigt ModelStress relaxation experiment: deformation at constant strain d/dt=0

dtdG +=

G=

This model can not describe stress relaxation !!!!


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More Complicated Models

Generalized Maxwell Model Voigt-Kelvin Model

..

{Gi, i}

{Gi, i}

These models describe systems with

multiple relaxation times.


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Generalized Maxwell Can

Describe Relaxation Process in Real Polymers

Th B lt S iti


34/43

The Boltzmann Superposition

PrincipleThe strain from any combination of small step stresses is the linear

combination of the strains resulting from each individual step

i

applied at time ti

(t)

(t)

1

2

2J(t-t2)t1 t2

1J(t-t1)


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The Boltzmann Superposition

Principle

For the multi-step loading program:

..)()()()( 332211 +++= ttJttJttJt

For continuously changing stress:

s

s

s

t

ss

t

s dtdt

tdttJtdttJt )()()()()(

==

For the strain experiments

ss

s

t

ss

t

s

dtdt

tdttGtdttGt

)()()()()(

==


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Creep and Recovery

(t)

(t)

0

t1 t2

r


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Dynamic Measurements

Elastic solid:

A sinusoidal strain with angular frequency

( )tt sin)( 0=

)sin()()( 0 tGtGt ==

The stress is perfectly in phase with the strain for Hookean solid.


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Dynamic Measurements

Newtonian liquid:

)2sin()cos(

)(

)( 00

+=== ttdttd

t

The stress is out of phase with the strain for Newtonian liquid.


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Dynamic MeasurementsIn general, the linear response of the viscoelastic materials always

has stress oscillate at the same frequency as the applied strain, but

the stress leads strain by aphase angle

)sin()( 0 += tt

The stress can be separated into two orthogonal functions that

oscillate with the same frequency, one in phase with the strain and

the other out-of-phase with the strain by /2

)cos()()sin()()( "'0 tGtGt +=

G() is the storage modulus G() is the loss modulusExamples:

Elastic solids: G() =G and G()=0

Newtonian liquids: G() =0 and G()=


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Dynamic MeasurementsWe can rewrite storage and loss modulus in terms of phase angle

)sin()cos()sin()cos()sin( ttt +=+

)cos()(0

0'

=G

Using

and

[ ])sin()cos()sin()cos()sin()( 00 tttt +=+=

One obtains

)sin()(0

0"

=G

)(

)(

)tan( '

"

G

G

=

)cos()()sin()()( "'0 tGtGt +=


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Application to the Maxwell Model

+=

dt

d

Gdt

d 1

)sin()( 0 += tt( )tt sin)( 0=For the time dependent strain and stress

Substitution to the Maxwell model

results in

)sin()cos(

1)cos( 000

+++=

tt

Gt

Collecting terms at cos(t) and sin (t) one has

)sin()cos(

1 000 +=

G

)cos()sin(

10 00 +=

G


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Application to the Maxwell Model

)()(

"' GG

G+=

)()(0

'" GG

G+=

Rewriting the last equations in terms of loss and storage modulus

Solving this system of equations for G and G one arrives at

22

22'

1)(

+= GG

22

"

1

)(

+

= GG

1

)(

)()(tan

'

"

==G

G

G

G


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Readings:

Ch. 3 p.p.51-66, M. T. Swah, W. J. MacKnight, Introduction toPolymer Viscoelasticity.

Ch. 7. p.p. 187-204 D. I. Bower, An Introduction to Polymer Physics

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Mechanical Properties Polymer_141013