Modulus vs. Viscosityy

Chapter 6 ~ 8 p

Mechanical Properties

Viscoelasticityy

(Chapter 6~7)

Modulus vs. Viscosityy

• Mechanical behavior

– Hooke’s law (ideally time-indep.) Ideal elastic body

– Ideal elastic body

, F , l

σ = E ε σ = ε = ^Δ

• Relaxation behavior

, ,

A l

Relaxation behavior

- Newtonian liquid

ideal viscous body (ideally time dep ) - ideal viscous body (ideally time-dep.)

d ε l

σ η ε ε =

,

= = ^Δ dt l

σ η ε ε = = =

Young’s modulus as a function of T

Young s modulus as a function of T

Uniaxial tension test

Uniaxial tension test

What polymers may exhibit What polymers may exhibit…

• Time-dep. Response

N f t i th l f t

• Non-recovery of strain upon the removal of stress

• Non-linearity of response

• Large strains without fracture

• Anisotropic response Anisotropic response

Elastic response of a ideal Hookian body

Under tension or compression

σ = E ε

( )

3 1 2 K = E

Poisson's ratio: l

ν

( )

3 1 2− ν

Metal: 0.3~0.4 vol. increase of 1 1 dV

K = −V dp

Bulk modulus: rubber: 0.5 No vol. change

decrease

K V dp g

Under shear Under shear

Shear modulus: G

σ

=

θ

( )

2 1 G E

= ν

(

)

2 1+ν (1 ) _i 3

i

l p

E

ν σ ν

+ +

= i =1,2,3

E i , ,3

ν 1 G ^E

When ^,

ν = 2

G = 3 When

Linear Viscoelasticity Linear Viscoelasticity

• A perfect Newtonian liq p q

- ideal viscous body , d l dt l σ η ε ε =

= ε = ^Δ

• A viscoelastic body ^{G d}

dt σ = θ η + θ ε

dt

• Time-dep. Creep p p

Creep compliance

( ) ( )

( ) l l t l t l t

Under constant load…

3

1 2

1 2 3

( ) ( )

( ) l t ( ) l l t l t ( ) ( )

J t J J t J t

σ σ σ σ

= = + + = + +

asymptotic as t goes to infinity

instantaneous linear in time linear in time

• Stress relaxation

( ) t J t ( t ) J t ( t ) J t ( t )

ε Δ σ + Δ σ + Δ σ +

Under a fixed strain…

1 1 2 2 3 3

( ) t J t ( t ) J t ( t ) J t ( t ) ...

ε = Δ σ − + Δ σ − + Δ σ − +

∫

or

( ) t

J t ( t d ') '

ε σ

= ∫

−

• Maxwell model

d d

d d

s d

,

d d

d d

dt dt dt Edt ε ε

ε σ σ

= + = + η

and

s d s d

σ σ = = σ ε ε ε = +

At constant strain,

0 ₌ d σ σ ₊

( E )

At constant strain,

0

Edt ⁺ η

0

exp( t )

σ σ ^{= −} η

• Voigt-Kelvin model g

d and

E ε

σ = E ε η + , ε ε =

= ε

and σ σ σ =

+

σ = ε η + dt ε ε = = ε σ σ σ = +

0

1 exp( E ) σ t

ε = ^⎡ ⎢ ^⎤ ⎥

Under constant load

σ σ = 1 exp( t )

ε E

= ⎢ − − η ⎥

⎣ ⎦

Under constant load,

σ σ =

• Dynamic measurements y

-complex modulus

[ ]

0

exp( iwt ) and

exp ( i wt )

ε ε = σ σ = + δ

[ ] [ ]

0 0

σ σ

complex modulus

σ

[ ] [ ]

0 0

1 2

0 0

* exp cos sin

G G iG σ σ i δ σ δ i δ

ε ε ε

= + = = = +

storage modulus storage modulus

loss modulus

Energy dissipated per cycle: ²

E π ε G

Δ =

-complex compliance p p

By definition, complex creep compliance

1 2

* 1 / *

J = G = J − iJ

1 2

1 2 2 2 2 2

1 2 1 2

and

G G

J J

G G G G

= =

+ +

1 2 1 2

0

exp( iwt ) ( G

iG e

)

σ σ = = +

Insertion of

Following Maxwell model,

2 2

1 2 2

and

1 1

Ew Ew

G G

w w

τ τ

τ τ

= =

+ +

1 + w τ 1 + w τ

Calculated results

Calculated results

• Time-temperature superposition p p p

• Glass transition in amorphous polymers p p y

• Factors affecting Tg -Main chain flexibility -Side groupSide group

-Plasticizers etc…

Yield and Fracture of Polymers y

(Chapter 8)

I t f Yi ld F t

Importance of Yield or Fracture

Yield Yield

Ideal yield behavior

Fracture

Fracture