Viscoelastic Damping: Classical Models

Viscoelastic Damping
Classical Models

Mohammad Tawfik

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http://WikiCourses.WikiSpaces.com

Objectives
• Recognize the nature of viscoelastic
material
• Understand the damping models of
viscoelastic material
• Dynamics of structures with viscoelastic
material

Mohammad Tawfik

WikiCourses#

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

Mohammad Tawfik

WikiCourses#

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:
Mohammad Tawfik

ε =ε s + ε d
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Stress-Strain Relation
• The stress is equal in
both elements and is
given by the relation:
• From which we may
write:
• Or:
Mohammad Tawfik

σ =E s ε s =C d ε d
˙
σ
σ
ε s = , ε d =∫ dt
Es
Cd

σ
σ
σ σ
˙
ε = +∫ dt ∧ ε = +
˙
Es
Cd
Es Cd
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Three Main Characteristics
• Creep
Strain changing with time for the same stress

• Relaxation
Stress changing with time for constant strain

• Storage and Loss Moduli
Effective modulus of elasticity in response to
frequency excitation
Mohammad Tawfik

WikiCourses#

Maxwell Model Characteristics
• Creep:
– For constant stress, we get:

σ
σ
˙
ε= +
˙
E s Cd

⏟
zero

– Which gives:

σ
ε= t
Cd

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

WikiCourses#

• Relaxation:
– For constant strain, we get:

σ σ
˙ +
0=
Es Cd

– Which gives:

−tE s /C d

σ =σ 0 e

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

WikiCourses#

• Storage and Loss Factors:

σ =σ 0 e
ε=ε 0 e

– For harmonic stress:
– Which drives the strain
harmonically:
– Giving:

(

)

jω 1
j ωε o =
+
σo
E s Cd
Mohammad Tawfik

σ o=

jωt
jωt

E s C d jω
E s + jωC d

εo

WikiCourses#

C
σ o=

(

σ o=

2

d2

Es ω + E
E

C
E

s

d

2

2+ ω
s

E s ω2
2

2 +ω

2

C

d

2C d
s

C

2

εo

d2

E
+j

jω

E

s

2Cd
s

2 +ω

2

ω
C

d

2

)

εo

'

σ o= E ( 1+ jη ) ε o
Mohammad Tawfik

WikiCourses#

Storage and Loss Moduli
• The stress strain relation of the
'
viscoelastic material appears to σ o= E ( 1+ jη ) ε o
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
Mohammad Tawfik

WikiCourses#

Frequency Dependent Behavior
1
0.9
0.8
0.7

M o d u lu s

0.6

E

0.5

u

0.4
0.3
0.2
0.1
0

0

2

4

6

8

10

Frequency

Mohammad Tawfik

WikiCourses#

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!

Mohammad Tawfik

WikiCourses#

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:
Mohammad Tawfik

σ =σ s + σ d
σ =E s ε s +C d ε d
˙
WikiCourses#

Kalvin-Voigt Model Characteristics
• Creep:
– For constant stress, we get:

−E s t /C d
σ
)
ε= (1−e
Es

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

Mohammad Tawfik

WikiCourses#

• Relaxation:
– For constant strain, we get:

σ =E s ε 0

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

WikiCourses#

Creep Relaxation Summary
Maxwell

Kalvin-Voigt

Creep

Bad

Good

Relaxation

Good

Bad

Mohammad Tawfik

WikiCourses#

• Storage and Loss Factors:

σ =σ 0 e
ε=ε 0 e

– For harmonic stress:
– Which drives the strain
harmonically:
– Giving:

jωt
jωt

σ =( E s + jωC d ) ε o
Mohammad Tawfik

WikiCourses#

Frequency Dependent Behavior
14
12

Modulus

10
8
6
E

4

u

2
0
0

2

4

6

8

10

Frequency

Mohammad Tawfik

WikiCourses#

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!

Mohammad Tawfik

WikiCourses#

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

Mohammad Tawfik

WikiCourses#

Viscoelastic Damping: Classical Models

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Viscoelastic Damping: Classical Models