What is viscoelastic damping?
What are the classical models?
What is creep?
What is Relaxation?
What is complex modulus?
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2. Objectives
• Recognize the nature of viscoelastic
material
• Understand the damping models of
viscoelastic material
• Dynamics of structures with viscoelastic
material
Viscoelastic Damping
Mohammad Tawfik
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3. 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
Viscoelastic Damping
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4. 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:
Viscoelastic Damping
Mohammad Tawfik
ε =ε s + ε d
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5. Stress-Strain Relation
• The stress is equal in
both elements and is
given by the relation:
• From which we may
write:
• Or:
Viscoelastic Damping
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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6. 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
Viscoelastic Damping
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7. 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!
Viscoelastic Damping
Mohammad Tawfik
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8. Maxwell Model Characteristics
• 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
Viscoelastic Damping
Mohammad Tawfik
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9. Maxwell Model Characteristics
• 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
Viscoelastic Damping
Mohammad Tawfik
σ o=
jωt
jωt
E s C d jω
E s + jωC d
εo
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10. Maxwell Model Characteristics
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
Viscoelastic Damping
Mohammad Tawfik
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11. 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
Viscoelastic Damping
Mohammad Tawfik
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12. 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
Viscoelastic Damping
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13. 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!
Viscoelastic Damping
Mohammad Tawfik
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14. 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:
Viscoelastic Damping
Mohammad Tawfik
σ =σ s + σ d
σ =E s ε s +C d ε d
˙
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15. 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!
Viscoelastic Damping
Mohammad Tawfik
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16. Kalvin-Voigt Model Characteristics
• 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!
Viscoelastic Damping
Mohammad Tawfik
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18. Kalvin-Voigt Model Characteristics
• 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
Viscoelastic Damping
Mohammad Tawfik
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20. 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!
Viscoelastic Damping
Mohammad Tawfik
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21. Assignment
• Study the creep, relaxation, and frequency
response characteristics of the Zener
model shown in the following sketch
Viscoelastic Damping
Mohammad Tawfik
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