Viscoelastic Damping: Zener model

Viscoelastic DampingMohammad Tawfik

#WikiCourseshttp://WikiCourses.WikiSpaces.com

Viscoelastic Damping

Zener Model



Zener Model

• The Zener model describes the material as a viscous damper in parallel with an elastic stiffness and both are in series with another stiffness.

• The strain may be written as: ε=ε s+ε1



Stress-Strain Relation

• The stress is is given by the relation:

• From which we may write:

• Or:

σ=E s ε s=E p ε1+Cd ε̇1

ε s=σEs, ε1=

σE p+sCd

ε=σE s

+σE p+sCd

¿σ(E p+sC d+E sE s (E p+sCd ) )



Stress-Strain Relation

• Giving:

• Back to time domain:

E s(E p+sC d ) ε=(E p+sC d+E s )σ

E sE pε+EsC d ε̇=(E p+E s) σ+Cd σ̇

Eε+Eβ ε̇=σ+α σ̇



Zener Model Characteristics

• Creep:– For constant stress, we get:

– Giving:

Eε+Eβ ε̇=σ0

ε=σ0E

−e−t / β

E s




• Relaxation:– For constant strain, we get:

– Which gives:

σ=σ 0+Eε0 (1−e−t /α )

Eε=σ+α σ̇




• Storage and Loss Factors:– For harmonic stress:– Which drives the strain

harmonically:– Giving:Eεo+ jωE βεo=σo+ j ωασo

σ=σ 0ejωt

ε=ε 0ejωt

σ o=E1+ j ωβ1+ j ωα

ε o=E1+ω2αβ+ jω ( β−α)

1+ω2α 2εo




σ o=E' (1+ jη ) εo

σ o=E(1+ω2αβ

1+ω2α2+jω ( β−α)1+ω2 α2 )εo



Frequency Dependent Behavior

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4

Frequency

Modulus

E

u



Other Models

• Some, more accurate, models were developed to represent the behavior of viscoelastic material

• The greatest concern was paid for the modeling in the time domain.

• The most famous models are:– Golla-Hughes-McTavish – Augmented Temperature Field– Fractional Derivative

Education

Viscoelastic Damping: Zener model