Viscoelastic Damping: Classical Models

Viscoelastic DampingMohammad Tawfik

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Viscoelastic Damping

Classical Models



Objectives

• Recognize the nature of viscoelastic material

• Understand the damping models of viscoelastic material

• Dynamics of structures with viscoelastic material



What is Viscoelastic Material?

• Materials that Exhibit, both, viscous and elastic characteristics.

• The material may be modeled in many different ways. Classical models include:– Mawxell Model– Kalvin-Voight Model



Maxwell Model

• The Maxwell model describes the material as a viscous damper in series with an elastic stiffness.

• When stress is applied, it is uniform through the element.

• The strain may be written as:

ε=ε s+εd



Stress-Strain Relation

• The stress is equal in both elements and is given by the relation:

• From which we may write:

• Or:

σ=E s ε s=Cd ε̇d

ε s=σE s, εd=∫

σCddt

ε=σE s

+∫σCddt ∧ ε̇=

σ̇E s

+σC d



Three Main Characteristics

• CreepStrain changing with time for the same stress

• RelaxationStress changing with time for constant strain

• Storage and Loss ModuliEffective modulus of elasticity in response to

frequency excitation



Maxwell Model Characteristics

• Creep:– For constant stress, we get:

– Which gives:

• Which indicates that the strain will grow to an unbound value as time increases!

ε̇=σ̇E s⏟zero

+σCd

ε=σCdt




• Relaxation:– For constant strain, we get:

– Which gives:

• Which means that the stress will decrease as time grows for the same strain

0=σ̇Es

+σCd

σ=σ 0e−tE

s/Cd




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

harmonically:– Giving:

j ωεo=( jωE s +1Cd )σ o

σ=σ 0ejωt

ε=ε 0ejωt

σ o=E sCd jω

E s+ jωC dεo




σ o=Cd 2E sω

2+E

s2C d jω

Es2+ω2C

d 2

εo

σ o=(Cd 2Esω

2

Es2+ω

2Cd2

+ jEs2Cd ω

Es2+ω

2Cd2 )εo

σ o=E' (1+ jη ) εo



Storage and Loss Moduli

• The stress strain relation of the viscoelastic material appears to contain a complex modulus of elasticity!

• The real part is called the storage modulus

• The imaginary part is called the loss modulus

• And their ratio is called the loss factor

σ o=E' (1+ jη ) εo



Frequency Dependent Behavior

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Frequency

Modulus E

u



Notes on the Maxwell Model

• Under static loading, the stiffness, storage modulus is zero, and the loss factor is infinity!

• For very high frequencies, the loss factor becomes zero!



Kalvin-Voigt Model

• The Kalvin-Voigt model describes the material as a viscous damper in parallel with an elastic stiffness.

• When stress is applied, it is distributed through the elements.

• The stress strain relation may be written as:

σ=σ s+σ d

σ=E sε s+C d ε̇d



Kalvin-Voigt Model Characteristics

• Creep:– For constant stress, we get:

• Which indicates that the strain will grow to a constant value as time increases!

ε=σEs

(1−e−E s t /C d)




• Relaxation:– For constant strain, we get:

• Which means that the stress will stay constant as time grows for the same strain!

σ=E sε 0



Creep Relaxation Summary

Maxwell Kalvin-Voigt

Creep Bad Good

Relaxation Good Bad




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

harmonically:– Giving:

σ=(E s+ jωC d ) εo

σ=σ 0ejωt

ε=ε 0ejωt



Frequency Dependent Behavior

0

2

4

6

8

10

12

14

0 2 4 6 8 10

Frequency

Mo

du

lus

E

u



Notes on the Kalvin-Voigt Model

• Under all loading, storage modulus is equal to the stiffness of the spring, and the loss factor is zero.

• For very high frequencies, the loss factor becomes grows unbound!



Assignment

• Study the creep, relaxation, and frequency response characteristics of the Zener model shown in the following sketch

Viscoelastic Damping: Classical Models

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