246448680 Linear Viscoelasticity

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Linear Viscoelasticity

description

vis

Transcript of 246448680 Linear Viscoelasticity

Linear Viscoelasticity

Elastic Response

Viscous Response

Maxwell Model

Creep

Stress Relaxation due to Maxwell

Voigt Model

Creep due to Voigt

Relaxation

Combination of Maxwell and Voigt

Burgers Model

Generalized Models

Continues Distribution: Maxwell

Continues Distribution: Voigt

Superposition Principle

Dynamic Response

Input:Viscoelastic

body

Output:

For a dashpot:

Stress: Moduli:

Complex Representation:

Time Scales

TTT(if it would be right….)

It is because

Master curve for Polymers

Composition of Relaxations: phase shift

Comparison of E(T) and E(t)

General Constitutive Law

))()((0 t

dtE

))()((1

0 t

dtKE

*)( KIE We can re-write this in the form:

*)( IE

If we define *)1( EE

than we generalize the Elastic law:

E

It might be shown that *)1(1

*1

11K

EE

and, for example:

)1(2

EG

Laplace Transform

Properties of Laplace Transform

Linear Viscoelasticity

0

0

0

zzzyzxz

yyzyyxy

xxzxyxx

Fzyx

Fzyx

Fzyx

)(2

1

i

j

j

iij x

u

x

u

ijijij 2

(no time, so far)

*)1(*);1(

Laplace…

;)()((0

dtt

;)1()(

ijijij )1(2)1(

Laplace transform of this function leads to

Similarly for

Finally:

)(2

1

i

j

j

iij x

u

x

u

0, jjij F

Examples of Operators

t

ctK )(

0

)(

t

ctK

10;)(

t

ctK

Boltzmann kernel

Boltzmann without singularity

No infinite rate of deformation

0

0ln1

t

cE

1

1

1t

c

E

Homework

• Pick a Linear viscoelastic moduli

• Solve the Lame problem